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In the formula for the ",{"__TSPROSE_proseElement":215,"schema":3844,"data":3845,"storageKey":3681,"id":3846},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":3680},"dependencyInliner-zZAa9vcG7-1",{"__TSPROSE_proseElement":215,"schema":3848,"data":3849},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", the middle term is ",{"__TSPROSE_proseElement":215,"schema":3851,"data":3852,"storageKey":3854,"id":3855},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3853},"2ab","$ 2ab $","inlinerMath-McBVEYBhA",{"__TSPROSE_proseElement":215,"schema":3857,"data":3858},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", so the job now is to rewrite ",{"__TSPROSE_proseElement":215,"schema":3860,"data":3861,"storageKey":3863,"id":3864},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3862},"40a^3b^4","$ 40a^3b^4 $","inlinerMath-bhAWJNQgL",{"__TSPROSE_proseElement":215,"schema":3866,"data":3867},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," in the form ",{"__TSPROSE_proseElement":215,"schema":3869,"data":3870,"storageKey":3854,"id":3871},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3853},"inlinerMath-McBVEYBhA-1",{"__TSPROSE_proseElement":215,"schema":3873,"data":3874},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," to see what ",{"__TSPROSE_proseElement":215,"schema":3876,"data":3877,"storageKey":3879,"id":3880},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3878},"b","$ b $","inlinerMath-LTS8nNQCZ",{"__TSPROSE_proseElement":215,"schema":3882,"data":3883},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," must be:","paragraph-865vikplq",{"__TSPROSE_proseElement":215,"schema":3886,"data":3887,"storageKey":3889,"id":3890},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3888,"freeze":219},"40a^3b^4 = 2 \\cdot 20a^3b^4 = 2 \\cdot \\underset{a}{(4a^3)} \\cdot \\underset{b}{(5b^4)}","$$ 40a^3b^4 = 2 \\cdot 20a^3b^4 = 2 \\cdot \\underset{a}{(4a^3)} \\cdot \\underset{b}{(5b^4)} $$","blockMath-3jkyrhaGD",{"__TSPROSE_proseElement":215,"schema":3892,"children":3893,"id":3913},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[3894,3897,3901,3904,3910],{"__TSPROSE_proseElement":215,"schema":3895,"data":3896},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"So in the square of a sum formula, the role of ",{"__TSPROSE_proseElement":215,"schema":3898,"data":3899,"storageKey":3879,"id":3900},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3878},"inlinerMath-LTS8nNQCZ-1",{"__TSPROSE_proseElement":215,"schema":3902,"data":3903},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," is played by ",{"__TSPROSE_proseElement":215,"schema":3905,"data":3906,"storageKey":3908,"id":3909},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3907},"5b^4","$ 5b^4 $","inlinerMath-OO73kGvNA",{"__TSPROSE_proseElement":215,"schema":3911,"data":3912},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Now we can write the expansion from the formula:","paragraph-ZpKu5niOR",{"__TSPROSE_proseElement":215,"schema":3915,"data":3916,"storageKey":3918,"id":3919},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3917,"freeze":219},"(4a^3 + 5b^4)^2 = (4a^3)^2 + 2 \\cdot (4a^3) \\cdot (5b^4) + (5b^4)^2 = 16a^6 + 40a^3b^4 + 25b^8","$$ (4a^3 + 5b^4)^2 = (4a^3)^2 + 2 \\cdot (4a^3) \\cdot (5b^4) + (5b^4)^2 = 16a^6 + 40a^3b^4 + 25b^8 $$","blockMath-jN84PjICZ",{"__TSPROSE_proseElement":215,"schema":3921,"data":3922,"children":3923},{"name":231,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[3924,3933,3939,3948],{"__TSPROSE_proseElement":215,"schema":3925,"children":3926},{"name":236,"type":218,"linkable":219,"__TSPROSE_schema":215},[3927],{"__TSPROSE_proseElement":215,"schema":3928,"data":3929,"storageKey":3931,"id":3932},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3930,"freeze":219},"(\\text{?} - 12a)(\\text{?} + \\text{?}) = 9b^2 - \\text{?}","$$ (\\text{?} - 12a)(\\text{?} + \\text{?}) = 9b^2 - \\text{?} $$","blockMath-kodHIw9Zk",{"__TSPROSE_proseElement":215,"schema":3934,"data":3935},{"name":257,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":3936},{"__ERUDIT_CHECK":215,"name":260,"data":3937},{"expr":3938},"9b^(2) - 144a^(2)",{"__TSPROSE_proseElement":215,"schema":3940,"children":3941},{"name":295,"type":218,"linkable":219,"__TSPROSE_schema":215},[3942],{"__TSPROSE_proseElement":215,"schema":3943,"data":3944,"storageKey":3946,"id":3947},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3945,"freeze":219},"(3b - 12a)(3b + 12a) = 9b^2 - 144a^2","$$ (3b - 12a)(3b + 12a) = 9b^2 - 144a^2 $$","blockMath-iSpdKYDhq",{"__TSPROSE_proseElement":215,"schema":3949,"children":3950},{"name":3659,"type":218,"linkable":219,"__TSPROSE_schema":215},[3951,3990,3996,4030],{"__TSPROSE_proseElement":215,"schema":3952,"children":3953,"id":3989},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[3954,3957,3962,3965,3971,3974,3980,3983,3987],{"__TSPROSE_proseElement":215,"schema":3955,"data":3956},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"The right-hand side is a ",{"__TSPROSE_proseElement":215,"schema":3958,"data":3959,"storageKey":3960,"id":3961},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquares","dependencyInliner-zJAa5vIur-1",{"__TSPROSE_proseElement":215,"schema":3963,"data":3964},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". By the difference of squares formula, the right-hand side equals ",{"__TSPROSE_proseElement":215,"schema":3966,"data":3967,"storageKey":3969,"id":3970},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3968},"a^2 - b^2","$ a^2 - b^2 $","inlinerMath-T1YOXnC6X",{"__TSPROSE_proseElement":215,"schema":3972,"data":3973},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", so ",{"__TSPROSE_proseElement":215,"schema":3975,"data":3976,"storageKey":3978,"id":3979},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3977},"9b^2","$ 9b^2 $","inlinerMath-oqv3m2PVW",{"__TSPROSE_proseElement":215,"schema":3981,"data":3982},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," is ",{"__TSPROSE_proseElement":215,"schema":3984,"data":3985,"storageKey":3773,"id":3986},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3772},"inlinerMath-eiXlIGTfO-1",{"__TSPROSE_proseElement":215,"schema":3988,"data":3694},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-2yDBUmDrO",{"__TSPROSE_proseElement":215,"schema":3991,"data":3992,"storageKey":3994,"id":3995},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3993,"freeze":219},"9b^2 = (3b)^2","$$ 9b^2 = (3b)^2 $$","blockMath-zvYWPdMun",{"__TSPROSE_proseElement":215,"schema":3997,"children":3998,"id":4029},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[3999,4002,4008,4011,4017,4020,4026],{"__TSPROSE_proseElement":215,"schema":4000,"data":4001},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"So the first unknown factor is ",{"__TSPROSE_proseElement":215,"schema":4003,"data":4004,"storageKey":4006,"id":4007},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":4005},"3b","$ 3b $","inlinerMath-OkRX6uD9W",{"__TSPROSE_proseElement":215,"schema":4009,"data":4010},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", and the second bracket is the matching one: ",{"__TSPROSE_proseElement":215,"schema":4012,"data":4013,"storageKey":4015,"id":4016},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":4014},"(3b + 12a)","$ (3b + 12a) $","inlinerMath-20552CXT0",{"__TSPROSE_proseElement":215,"schema":4018,"data":4019},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". 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Apply the formula, then expand with the ",{"__TSPROSE_proseElement":215,"schema":5698,"data":5699,"storageKey":3681,"id":5700},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":3680},"dependencyInliner-zZAa9vcG7-5",{"__TSPROSE_proseElement":215,"schema":5702,"data":4927},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-tJU29rsgZ",{"__TSPROSE_proseElement":215,"schema":5705,"data":5706,"storageKey":5708,"id":5709},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5707,"freeze":219},"((\\sqrt5 + \\sqrt6) + \\sqrt7)((\\sqrt5 + \\sqrt6) - \\sqrt7) = \\\\ (\\sqrt5 + \\sqrt6)^2 - (\\sqrt7)^2 = \\\\ (5 + 2 \\cdot \\sqrt{30} + 6) - 7 = \\\\ 4 + 2 \\cdot \\sqrt{30}","$$ ((\\sqrt5 + \\sqrt6) + \\sqrt7)((\\sqrt5 + \\sqrt6) - \\sqrt7) = \\\\ (\\sqrt5 + \\sqrt6)^2 - (\\sqrt7)^2 = \\\\ (5 + 2 \\cdot \\sqrt{30} + 6) - 7 = \\\\ 4 + 2 \\cdot \\sqrt{30} $$","blockMath-5xwqTjek3",{"__TSPROSE_proseElement":215,"schema":5711,"children":5712,"id":5716},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[5713],{"__TSPROSE_proseElement":215,"schema":5714,"data":5715},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"The two remaining brackets also form a difference of squares if we cleverly factor out a minus sign and reorder the terms. Apply the formula and expand everything we can:","paragraph-UQZlQop1E",{"__TSPROSE_proseElement":215,"schema":5718,"data":5719,"storageKey":5721,"id":5722},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5720,"freeze":219},"(\\sqrt5 - \\sqrt6 + \\sqrt7)(-\\sqrt5 + \\sqrt6 + \\sqrt7) = \\\\ (\\sqrt7 + (\\sqrt5 - \\sqrt6))(\\sqrt7 - (\\sqrt5 - \\sqrt6)) = \\\\ (\\sqrt7)^2 - (\\sqrt5 - \\sqrt6)^2 = \\\\ 7 - (5 - 2 \\cdot \\sqrt{30} + 6) = \\\\ -4 + 2 \\cdot \\sqrt{30}","$$ (\\sqrt5 - \\sqrt6 + \\sqrt7)(-\\sqrt5 + \\sqrt6 + \\sqrt7) = \\\\ (\\sqrt7 + (\\sqrt5 - \\sqrt6))(\\sqrt7 - (\\sqrt5 - \\sqrt6)) = \\\\ (\\sqrt7)^2 - (\\sqrt5 - \\sqrt6)^2 = \\\\ 7 - (5 - 2 \\cdot \\sqrt{30} + 6) = \\\\ -4 + 2 \\cdot \\sqrt{30} $$","blockMath-yQbj4cHri",{"__TSPROSE_proseElement":215,"schema":5724,"children":5725,"id":5729},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[5726],{"__TSPROSE_proseElement":215,"schema":5727,"data":5728},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Now multiply the two results:","paragraph-sMTfwaNBG",{"__TSPROSE_proseElement":215,"schema":5731,"data":5732,"storageKey":5734,"id":5735},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5733,"freeze":219},"(4 + 2 \\cdot \\sqrt{30})(-4 + 2 \\cdot \\sqrt{30}) = \\\\ (2 \\cdot \\sqrt{30} + 4)(2 \\cdot \\sqrt{30} - 4) = \\\\ (2 \\cdot \\sqrt{30})^2 - 4^2 = \\\\ 4 \\cdot 30 - 16 = \\\\ 104","$$ (4 + 2 \\cdot \\sqrt{30})(-4 + 2 \\cdot \\sqrt{30}) = \\\\ (2 \\cdot \\sqrt{30} + 4)(2 \\cdot \\sqrt{30} - 4) = \\\\ (2 \\cdot \\sqrt{30})^2 - 4^2 = \\\\ 4 \\cdot 30 - 16 = \\\\ 104 $$","blockMath-BLCOH1aUr","radical-mayhem",{"__TSPROSE_proseElement":215,"schema":5738,"data":5739,"children":5742,"id":6418},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":5740,"level":4657,"attributes":5741},"Time to Simplify",[],[5743,5750,5840,5935,6014,6087,6176,6291],{"__TSPROSE_proseElement":215,"schema":5744,"children":5745,"id":5749},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[5746],{"__TSPROSE_proseElement":215,"schema":5747,"data":5748},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Simplify the expression:","paragraph-rtI8tL6S7",{"__TSPROSE_proseElement":215,"schema":5751,"data":5752,"children":5753},{"name":231,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[5754,5763,5769,5778],{"__TSPROSE_proseElement":215,"schema":5755,"children":5756},{"name":236,"type":218,"linkable":219,"__TSPROSE_schema":215},[5757],{"__TSPROSE_proseElement":215,"schema":5758,"data":5759,"storageKey":5761,"id":5762},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5760,"freeze":219},"\\frac{4x^2 + 12x + 9}{4x^2 - 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- 1}{4^2} \\right) \\cdots \\left( \\frac{n^2 - 1}{n^2} \\right)","$$ \\left( \\frac{2^2 - 1}{2^2} \\right) \\left( \\frac{3^2 - 1}{3^2} \\right) \\left( \\frac{4^2 - 1}{4^2} \\right) \\cdots \\left( \\frac{n^2 - 1}{n^2} \\right) $$","blockMath-MO3dIqzjC",{"__TSPROSE_proseElement":215,"schema":6686,"children":6687,"id":6698},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6688,6691,6695],{"__TSPROSE_proseElement":215,"schema":6689,"data":6690},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"The numerators become a ",{"__TSPROSE_proseElement":215,"schema":6692,"data":6693,"storageKey":3960,"id":6694},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-20",{"__TSPROSE_proseElement":215,"schema":6696,"data":6697},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", which we rewrite as a 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After that, carry out large-scale cancellation, which gets rid of almost all the factors:","paragraph-uK7F24JN8",{"__TSPROSE_proseElement":215,"schema":6726,"data":6727,"storageKey":6729,"id":6730},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6728,"freeze":219},"\\frac{(\\cancel{1 \\cdot 2 \\cdot 3 \\cdots (n-1)})(\\cancel{3 \\cdot 4 \\cdot 5 \\cdots n} \\cdot (n+1))}{(2 \\cdot \\cancel{3 \\cdots (n-1)} \\cdot n) \\cdot (\\cancel{2 \\cdot 3 \\cdots n})} = \\frac{n+1}{2n}","$$ \\frac{(\\cancel{1 \\cdot 2 \\cdot 3 \\cdots (n-1)})(\\cancel{3 \\cdot 4 \\cdot 5 \\cdots n} \\cdot (n+1))}{(2 \\cdot \\cancel{3 \\cdots (n-1)} \\cdot n) \\cdot (\\cancel{2 \\cdot 3 \\cdots n})} = \\frac{n+1}{2n} $$","blockMath-9dqC5pAMr",{"__TSPROSE_proseElement":215,"schema":6732,"data":6586,"children":6733},{"name":6669,"type":218,"linkable":219,"__TSPROSE_schema":215},[6734,6748],{"__TSPROSE_proseElement":215,"schema":6735,"children":6736,"id":6747},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6737,6740,6744],{"__TSPROSE_proseElement":215,"schema":6738,"data":6739},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Substitute ",{"__TSPROSE_proseElement":215,"schema":6741,"data":6742,"storageKey":6652,"id":6743},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6651},"inlinerMath-jLscXzwad-1",{"__TSPROSE_proseElement":215,"schema":6745,"data":6746},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," into the simplified expression:","paragraph-gWVzocR9k",{"__TSPROSE_proseElement":215,"schema":6749,"data":6750,"storageKey":6752,"id":6753},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6751,"freeze":219},"\\frac{n+1}{2n} = \\frac{100+1}{2 \\cdot 100} = \\frac{101}{200}","$$ \\frac{n+1}{2n} = \\frac{100+1}{2 \\cdot 100} = \\frac{101}{200} $$","blockMath-JfWsJ7Ua7","fraction-snake",{"__TSPROSE_proseElement":215,"schema":6756,"data":6757,"children":6761,"id":6921},{"name":4653,"type":218,"linkable":215,"__TSPROSE_schema":215},{"info":6758},{"title":6759,"level":4657,"attributes":6760},"Difference of Exponents",[],[6762,6771,6776,6792,6808,6819],{"__TSPROSE_proseElement":215,"schema":6763,"children":6764},{"name":236,"type":218,"linkable":219,"__TSPROSE_schema":215},[6765],{"__TSPROSE_proseElement":215,"schema":6766,"data":6767,"storageKey":6769,"id":6770},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6768,"freeze":219},"2^x - 2^y = 1 >>{big} 4^x - 4^y = \\frac{5}{3} >>{big} x - y = \\text{?}","$$ 2^x - 2^y = 1 >>{big} 4^x - 4^y = \\frac{5}{3} >>{big} x - y = \\text{?} $$","blockMath-XfAYthEYJ",{"__TSPROSE_proseElement":215,"schema":6772,"data":6773},{"name":257,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":6774},{"__ERUDIT_CHECK":215,"name":4684,"data":6775},{"expr":5221},{"__TSPROSE_proseElement":215,"schema":6777,"children":6778},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[6779],{"__TSPROSE_proseElement":215,"schema":6780,"children":6781,"id":6791},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6782,6785,6789],{"__TSPROSE_proseElement":215,"schema":6783,"data":6784},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"In the second equality, play around with the powers and use the formula for the ",{"__TSPROSE_proseElement":215,"schema":6786,"data":6787,"storageKey":3960,"id":6788},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-21",{"__TSPROSE_proseElement":215,"schema":6790,"data":291},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-lpQQR3iMZ",{"__TSPROSE_proseElement":215,"schema":6793,"children":6794},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[6795,6802],{"__TSPROSE_proseElement":215,"schema":6796,"children":6797,"id":6801},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6798],{"__TSPROSE_proseElement":215,"schema":6799,"data":6800},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Use the following property of exponents:","paragraph-iMNb2Cf3a",{"__TSPROSE_proseElement":215,"schema":6803,"data":6804,"storageKey":6806,"id":6807},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6805,"freeze":219},"\\frac{a^{n}}{a^{m}} = a^{n-m}","$$ \\frac{a^{n}}{a^{m}} = a^{n-m} $$","blockMath-Mqz3eIPHQ",{"__TSPROSE_proseElement":215,"schema":6809,"children":6810},{"name":295,"type":218,"linkable":219,"__TSPROSE_schema":215},[6811],{"__TSPROSE_proseElement":215,"schema":6812,"children":6813,"id":6818},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6814],{"__TSPROSE_proseElement":215,"schema":6815,"data":6816,"storageKey":5222,"id":6817},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":5221},"inlinerMath-z9Yar9waf-1","paragraph-Z7KWRn2Ap",{"__TSPROSE_proseElement":215,"schema":6820,"children":6821},{"name":3659,"type":218,"linkable":219,"__TSPROSE_schema":215},[6822,6829,6835,6848,6854,6861,6867,6874,6880,6887,6893,6915],{"__TSPROSE_proseElement":215,"schema":6823,"children":6824,"id":6828},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6825],{"__TSPROSE_proseElement":215,"schema":6826,"data":6827},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"In the second equality, rewrite the fours as squared twos and play around with the exponents:","paragraph-1tEAXnWO1",{"__TSPROSE_proseElement":215,"schema":6830,"data":6831,"storageKey":6833,"id":6834},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6832,"freeze":219},"\\frac{5}{3} = 4^x - 4^y = (2^2)^x - (2^2)^y = 2^{2x} - 2^{2y} = (2^x)^2 - (2^y)^2","$$ \\frac{5}{3} = 4^x - 4^y = (2^2)^x - (2^2)^y = 2^{2x} - 2^{2y} = (2^x)^2 - (2^y)^2 $$","blockMath-tVQY9rIex",{"__TSPROSE_proseElement":215,"schema":6836,"children":6837,"id":6847},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6838,6841,6845],{"__TSPROSE_proseElement":215,"schema":6839,"data":6840},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"We get a ",{"__TSPROSE_proseElement":215,"schema":6842,"data":6843,"storageKey":3960,"id":6844},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-22",{"__TSPROSE_proseElement":215,"schema":6846,"data":4028},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-OARDI8FLE",{"__TSPROSE_proseElement":215,"schema":6849,"data":6850,"storageKey":6852,"id":6853},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6851,"freeze":219},"\\frac{5}{3} = (2^x)^2 - (2^y)^2 = (2^x + 2^y)\\underbrace{(2^x - 2^y)}_{\\small 1} \\\\ 2^x + 2^y = \\frac{5}{3} \\\\","$$ \\frac{5}{3} = (2^x)^2 - (2^y)^2 = (2^x + 2^y)\\underbrace{(2^x - 2^y)}_{\\small 1} \\\\ 2^x + 2^y = \\frac{5}{3} \\\\ $$","blockMath-ec7F6GpJM",{"__TSPROSE_proseElement":215,"schema":6855,"children":6856,"id":6860},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6857],{"__TSPROSE_proseElement":215,"schema":6858,"data":6859},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Now consider two sums. Here is the first:","paragraph-pBOiH7wof",{"__TSPROSE_proseElement":215,"schema":6862,"data":6863,"storageKey":6865,"id":6866},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6864,"freeze":219},"(2^x + \\cancel{2^y}) + (2^x - \\cancel{2^y}) = \\frac{5}{3} + 1 \\\\ 2 \\cdot 2^x = \\frac{8}{3} \\\\ 2^x = \\frac{4}{3}","$$ (2^x + \\cancel{2^y}) + (2^x - \\cancel{2^y}) = \\frac{5}{3} + 1 \\\\ 2 \\cdot 2^x = \\frac{8}{3} \\\\ 2^x = \\frac{4}{3} $$","blockMath-bNuaRNc8p",{"__TSPROSE_proseElement":215,"schema":6868,"children":6869,"id":6873},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6870],{"__TSPROSE_proseElement":215,"schema":6871,"data":6872},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"And here is the second:","paragraph-IWD1WIjDO",{"__TSPROSE_proseElement":215,"schema":6875,"data":6876,"storageKey":6878,"id":6879},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6877,"freeze":219},"(2^x + \\cancel{2^y}) - (2^x - \\cancel{2^y}) = \\frac{5}{3} - 1 \\\\ 2 \\cdot 2^y = \\frac{2}{3} \\\\ 2^y = \\frac{1}{3}","$$ (2^x + \\cancel{2^y}) - (2^x - \\cancel{2^y}) = \\frac{5}{3} - 1 \\\\ 2 \\cdot 2^y = \\frac{2}{3} \\\\ 2^y = \\frac{1}{3} $$","blockMath-kEHPqeJLA",{"__TSPROSE_proseElement":215,"schema":6881,"children":6882,"id":6886},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6883],{"__TSPROSE_proseElement":215,"schema":6884,"data":6885},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Division of powers with the same base can be written through the difference of their exponents:","paragraph-s2bb4WBtE",{"__TSPROSE_proseElement":215,"schema":6888,"data":6889,"storageKey":6891,"id":6892},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6890,"freeze":219},"\\frac{2^x}{2^y} = 2^{x-y} = \\frac{\\frac{4}{3}}{\\frac{1}{3}} = 4 = 2^2 \\\\ 2^{x-y} = 2^2 \\\\","$$ \\frac{2^x}{2^y} = 2^{x-y} = \\frac{\\frac{4}{3}}{\\frac{1}{3}} = 4 = 2^2 \\\\ 2^{x-y} = 2^2 \\\\ $$","blockMath-yEbfyiPFc",{"__TSPROSE_proseElement":215,"schema":6894,"children":6895,"id":6914},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6896,6899,6905,6908,6912],{"__TSPROSE_proseElement":215,"schema":6897,"data":6898},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"For the equality to hold, the difference ",{"__TSPROSE_proseElement":215,"schema":6900,"data":6901,"storageKey":6903,"id":6904},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6902},"x-y","$ x-y $","inlinerMath-9jMKf5UEV",{"__TSPROSE_proseElement":215,"schema":6906,"data":6907},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," must equal ",{"__TSPROSE_proseElement":215,"schema":6909,"data":6910,"storageKey":5222,"id":6911},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":5221},"inlinerMath-z9Yar9waf-2",{"__TSPROSE_proseElement":215,"schema":6913,"data":4028},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-VD4g7nocP",{"__TSPROSE_proseElement":215,"schema":6916,"data":6917,"storageKey":6919,"id":6920},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6918,"freeze":219},"x - y = 2","$$ x - y = 2 $$","blockMath-ugpdcuZjk","difference-of-exponents",{"__TSPROSE_proseElement":215,"schema":6923,"data":6924,"children":6928,"id":7228},{"name":4653,"type":218,"linkable":215,"__TSPROSE_schema":215},{"info":6925},{"title":6926,"level":4657,"attributes":6927},"The Mystery of 693",[6543],[6929,6948,6962,7002,7023],{"__TSPROSE_proseElement":215,"schema":6930,"children":6931},{"name":236,"type":218,"linkable":219,"__TSPROSE_schema":215},[6932],{"__TSPROSE_proseElement":215,"schema":6933,"children":6934,"id":6947},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6935,6938,6944],{"__TSPROSE_proseElement":215,"schema":6936,"data":6937},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"The difference of squares of two two-digit numbers written with the same digits equals ",{"__TSPROSE_proseElement":215,"schema":6939,"data":6940,"storageKey":6942,"id":6943},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6941},"693","$ 693 $","inlinerMath-jtkSj1W4b",{"__TSPROSE_proseElement":215,"schema":6945,"data":6946},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Find those numbers.","paragraph-Uu7Xa49fs",{"__TSPROSE_proseElement":215,"schema":6949,"data":6950},{"name":257,"type":218,"linkable":219,"__TSPROSE_schema":215},{"label":6951,"serializedValidator":6952},"Two numbers written with the same digits",{"type":6953,"ordered":219,"separator":6954,"answers":6955},"array",",",[6956,6959],{"__ERUDIT_CHECK":215,"name":4684,"data":6957},{"expr":6958},"43",{"__ERUDIT_CHECK":215,"name":4684,"data":6960},{"expr":6961},"34",{"__TSPROSE_proseElement":215,"schema":6963,"children":6964},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[6965,6996],{"__TSPROSE_proseElement":215,"schema":6966,"children":6967,"id":6995},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[6968,6971,6975,6977,6983,6986,6992],{"__TSPROSE_proseElement":215,"schema":6969,"data":6970},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"A two-digit number written with digits ",{"__TSPROSE_proseElement":215,"schema":6972,"data":6973,"storageKey":5091,"id":6974},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":5090},"inlinerMath-hbA4JBEYl-3",{"__TSPROSE_proseElement":215,"schema":6976,"data":4788},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":6978,"data":6979,"storageKey":6981,"id":6982},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6980},"m","$ m $","inlinerMath-uuvX6OFxQ",{"__TSPROSE_proseElement":215,"schema":6984,"data":6985},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," looks like this: ",{"__TSPROSE_proseElement":215,"schema":6987,"data":6988,"storageKey":6990,"id":6991},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6989},"10 \\cdot n + m","$ 10 \\cdot n + m $","inlinerMath-uk9j6GD1a",{"__TSPROSE_proseElement":215,"schema":6993,"data":6994},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Examples:","paragraph-AiXVaWxrg",{"__TSPROSE_proseElement":215,"schema":6997,"data":6998,"storageKey":7000,"id":7001},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":6999,"freeze":219},"59 = 10 \\cdot 5 + 9 >>{big} 13 = 10 \\cdot 1 + 3 >>{big} 92 = 10 \\cdot 9 + 2 >>{big}","$$ 59 = 10 \\cdot 5 + 9 >>{big} 13 = 10 \\cdot 1 + 3 >>{big} 92 = 10 \\cdot 9 + 2 >>{big} $$","blockMath-EwD1ai13c",{"__TSPROSE_proseElement":215,"schema":7003,"children":7004},{"name":295,"type":218,"linkable":219,"__TSPROSE_schema":215},[7005],{"__TSPROSE_proseElement":215,"schema":7006,"children":7007,"id":7022},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[7008,7013,7015,7020],{"__TSPROSE_proseElement":215,"schema":7009,"data":7010,"storageKey":7011,"id":7012},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6958},"$ 43 $","inlinerMath-WIFfESNXa",{"__TSPROSE_proseElement":215,"schema":7014,"data":4788},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":7016,"data":7017,"storageKey":7018,"id":7019},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6961},"$ 34 $","inlinerMath-ckxzaGhnI",{"__TSPROSE_proseElement":215,"schema":7021,"data":291},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-eGlfCyjFQ",{"__TSPROSE_proseElement":215,"schema":7024,"children":7025},{"name":3659,"type":218,"linkable":219,"__TSPROSE_schema":215},[7026,7076,7082,7098,7104,7150,7156,7209],{"__TSPROSE_proseElement":215,"schema":7027,"children":7028,"id":7075},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[7029,7032,7036,7038,7042,7045,7051,7053,7059,7062,7066,7069,7073],{"__TSPROSE_proseElement":215,"schema":7030,"data":7031},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Let the two unknown digits making up both two-digit numbers be ",{"__TSPROSE_proseElement":215,"schema":7033,"data":7034,"storageKey":5091,"id":7035},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":5090},"inlinerMath-hbA4JBEYl-4",{"__TSPROSE_proseElement":215,"schema":7037,"data":4788},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":7039,"data":7040,"storageKey":6981,"id":7041},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6980},"inlinerMath-uuvX6OFxQ-1",{"__TSPROSE_proseElement":215,"schema":7043,"data":7044},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Then the two numbers can be written as ",{"__TSPROSE_proseElement":215,"schema":7046,"data":7047,"storageKey":7049,"id":7050},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7048},"10n + m","$ 10n + m $","inlinerMath-Ay7NwYJLA",{"__TSPROSE_proseElement":215,"schema":7052,"data":4788},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":7054,"data":7055,"storageKey":7057,"id":7058},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7056},"10m + n","$ 10m + n $","inlinerMath-koNFoiBF2",{"__TSPROSE_proseElement":215,"schema":7060,"data":7061},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Write their ",{"__TSPROSE_proseElement":215,"schema":7063,"data":7064,"storageKey":3960,"id":7065},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-23",{"__TSPROSE_proseElement":215,"schema":7067,"data":7068},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", which by the condition equals ",{"__TSPROSE_proseElement":215,"schema":7070,"data":7071,"storageKey":6942,"id":7072},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6941},"inlinerMath-jtkSj1W4b-1",{"__TSPROSE_proseElement":215,"schema":7074,"data":4028},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-CK9pMOtp4",{"__TSPROSE_proseElement":215,"schema":7077,"data":7078,"storageKey":7080,"id":7081},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":7079,"freeze":219},"(10n + m)^2 - (10m + n)^2 = 693 \\\\ (10n + m + 10m + n)(10n + m - 10m - n) = 693 \\\\ (11n + 11m)(9n - 9m) = 693 \\\\ 99(n + m)(n - m) = 693","$$ (10n + m)^2 - (10m + n)^2 = 693 \\\\ (10n + m + 10m + n)(10n + m - 10m - n) = 693 \\\\ (11n + 11m)(9n - 9m) = 693 \\\\ 99(n + m)(n - m) = 693 $$","blockMath-EYhnSyp7Q",{"__TSPROSE_proseElement":215,"schema":7083,"children":7084,"id":7097},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[7085,7088,7094],{"__TSPROSE_proseElement":215,"schema":7086,"data":7087},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"On the left, we get a product of factors, one of which is ",{"__TSPROSE_proseElement":215,"schema":7089,"data":7090,"storageKey":7092,"id":7093},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7091},"99","$ 99 $","inlinerMath-XT2eVJO0l",{"__TSPROSE_proseElement":215,"schema":7095,"data":7096},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Get rid of it so we can work with simpler numbers.","paragraph-4O1f4EhHU",{"__TSPROSE_proseElement":215,"schema":7099,"data":7100,"storageKey":7102,"id":7103},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":7101,"freeze":219},"(n + m)(n - m) = \\frac{693}{99} = 7 \\\\ (n + m)(n - m) = 7","$$ (n + m)(n - m) = \\frac{693}{99} = 7 \\\\ (n + m)(n - m) = 7 $$","blockMath-kCZjCWfpu",{"__TSPROSE_proseElement":215,"schema":7105,"children":7106,"id":7149},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[7107,7110,7116,7118,7124,7127,7133,7136,7140,7142,7146],{"__TSPROSE_proseElement":215,"schema":7108,"data":7109},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"So we have this situation: two natural numbers ",{"__TSPROSE_proseElement":215,"schema":7111,"data":7112,"storageKey":7114,"id":7115},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7113},"n+m","$ n+m $","inlinerMath-BFYof9u2B",{"__TSPROSE_proseElement":215,"schema":7117,"data":4788},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":7119,"data":7120,"storageKey":7122,"id":7123},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7121},"n-m","$ n-m $","inlinerMath-VjYEJ76iR",{"__TSPROSE_proseElement":215,"schema":7125,"data":7126},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," multiply to give ",{"__TSPROSE_proseElement":215,"schema":7128,"data":7129,"storageKey":7131,"id":7132},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7130},"7","$ 7 $","inlinerMath-sYh3uEvdQ",{"__TSPROSE_proseElement":215,"schema":7134,"data":7135},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". 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From the explicit data, yes, but any number can be multiplied by ",{"__TSPROSE_proseElement":215,"schema":8240,"data":8241,"storageKey":4744,"id":8242},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":4743},"inlinerMath-GWJ3eIxvK-6",{"__TSPROSE_proseElement":215,"schema":8244,"data":8245},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," without changing it:","paragraph-00TtYgJZa",{"__TSPROSE_proseElement":215,"schema":8248,"data":8249,"storageKey":8251,"id":8252},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8250,"freeze":219},"\\ldots = \\left[ (k+1) + k \\right] \\cdot \\brand{1} = \\ldots","$$ \\ldots = \\left[ (k+1) + k \\right] \\cdot \\brand{1} = \\ldots $$","blockMath-C2F1mwlx8",{"__TSPROSE_proseElement":215,"schema":8254,"children":8255,"id":8266},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8256,8259,8263],{"__TSPROSE_proseElement":215,"schema":8257,"data":8258},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Now that one must be written as a bracket that simplifies to one. To do that, repeat the left bracket, but subtract ",{"__TSPROSE_proseElement":215,"schema":8260,"data":8261,"storageKey":8163,"id":8262},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8162},"inlinerMath-KaVbgkHhg-1",{"__TSPROSE_proseElement":215,"schema":8264,"data":8265},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," this time:","paragraph-eU5hKEN70",{"__TSPROSE_proseElement":215,"schema":8268,"data":8269,"storageKey":8271,"id":8272},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8270,"freeze":219},"\\ldots = \\left[ (k+1) + k \\right] \\cdot \\underbrace{\\left[ (k+1) - k \\right]}_{\\small 1}","$$ \\ldots = \\left[ (k+1) + k \\right] \\cdot \\underbrace{\\left[ (k+1) - k \\right]}_{\\small 1} $$","blockMath-WSr7q8jZQ",{"__TSPROSE_proseElement":215,"schema":8274,"children":8275,"id":8279},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8276],{"__TSPROSE_proseElement":215,"schema":8277,"data":8278},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"We got two identical brackets that differ only by the sign between the terms. Apply the difference-of-squares formula:","paragraph-lzmgFPgCp",{"__TSPROSE_proseElement":215,"schema":8281,"data":8282,"storageKey":8284,"id":8285},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8283,"freeze":219},"2k + 1 = \\left[ (k+1) + k \\right] \\cdot \\left[ (k+1) - k \\right] = (k+1)^2 - k^2 \\\\ 2k + 1 = (k+1)^2 - k^2","$$ 2k + 1 = \\left[ (k+1) + k \\right] \\cdot \\left[ (k+1) - k \\right] = (k+1)^2 - k^2 \\\\ 2k + 1 = (k+1)^2 - k^2 $$","blockMath-2yB364xXk",{"__TSPROSE_proseElement":215,"schema":8287,"children":8288,"id":8293},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8289],{"__TSPROSE_proseElement":215,"schema":8290,"data":8291,"storageKey":4857,"id":8292},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":4856},"inlinerMath-77K6blkcr-4","paragraph-ea37qO3lg-4",{"__TSPROSE_proseElement":215,"schema":8295},{"name":8296,"type":218,"linkable":219,"__TSPROSE_schema":215},"hr",{"__TSPROSE_proseElement":215,"schema":8298,"children":8299,"id":8342},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8300,8303,8307,8309,8313,8315,8319,8322,8326,8329,8333,8336,8340],{"__TSPROSE_proseElement":215,"schema":8301,"data":8302},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Now let us write the difference of squares for the numbers ",{"__TSPROSE_proseElement":215,"schema":8304,"data":8305,"storageKey":7131,"id":8306},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":7130},"inlinerMath-sYh3uEvdQ-3",{"__TSPROSE_proseElement":215,"schema":8308,"data":8061},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":8310,"data":8311,"storageKey":8066,"id":8312},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8065},"inlinerMath-eEHhsOp1e-1",{"__TSPROSE_proseElement":215,"schema":8314,"data":8070},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":8316,"data":8317,"storageKey":8075,"id":8318},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8074},"inlinerMath-b1yiDnYmR-1",{"__TSPROSE_proseElement":215,"schema":8320,"data":8321},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". To do that, write all these numbers as ",{"__TSPROSE_proseElement":215,"schema":8323,"data":8324,"storageKey":8128,"id":8325},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8127},"inlinerMath-yWT1AujDY-3",{"__TSPROSE_proseElement":215,"schema":8327,"data":8328},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", because we need to find ",{"__TSPROSE_proseElement":215,"schema":8330,"data":8331,"storageKey":8163,"id":8332},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8162},"inlinerMath-KaVbgkHhg-2",{"__TSPROSE_proseElement":215,"schema":8334,"data":8335},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," for each number. Subtract one from each number and divide by ",{"__TSPROSE_proseElement":215,"schema":8337,"data":8338,"storageKey":5222,"id":8339},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":5221},"inlinerMath-z9Yar9waf-4",{"__TSPROSE_proseElement":215,"schema":8341,"data":4028},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-8G1vcgHZ4",{"__TSPROSE_proseElement":215,"schema":8344,"data":8345,"storageKey":8347,"id":8348},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8346,"freeze":219},"7 = 2 \\cdot 3 + 1 \\Rightarrow 3 >>{big} 111 = 2 \\cdot 55 + 1 \\Rightarrow 55 >>{big} 507 = 2 \\cdot 253 + 1 \\Rightarrow 253","$$ 7 = 2 \\cdot 3 + 1 \\Rightarrow 3 >>{big} 111 = 2 \\cdot 55 + 1 \\Rightarrow 55 >>{big} 507 = 2 \\cdot 253 + 1 \\Rightarrow 253 $$","blockMath-blIsrvaI5",{"__TSPROSE_proseElement":215,"schema":8350,"children":8351,"id":8362},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8352,8355,8359],{"__TSPROSE_proseElement":215,"schema":8353,"data":8354},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Use the values of ",{"__TSPROSE_proseElement":215,"schema":8356,"data":8357,"storageKey":8163,"id":8358},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8162},"inlinerMath-KaVbgkHhg-3",{"__TSPROSE_proseElement":215,"schema":8360,"data":8361},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," we found to write the difference of squares:","paragraph-rvgQR3ayz",{"__TSPROSE_proseElement":215,"schema":8364,"data":8365,"storageKey":8367,"id":8368},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8366,"freeze":219},"7 = \\boxed{4^2 - 3^2} >>{big} 111 = \\boxed{56^2 - 55^2} >>{big} 507 = \\boxed{254^2 - 253^2}","$$ 7 = \\boxed{4^2 - 3^2} >>{big} 111 = \\boxed{56^2 - 55^2} >>{big} 507 = \\boxed{254^2 - 253^2} $$","blockMath-UuLlIqHHc","any-odd-is-a-difference-of-squares",{"__TSPROSE_proseElement":215,"schema":8371,"data":8372,"children":8376,"id":8609},{"name":4653,"type":218,"linkable":215,"__TSPROSE_schema":215},{"info":8373},{"title":8374,"level":8036,"attributes":8375},"Factoring the Cube of a Sum and a Difference",[],[8377,8393,8403,8433,8482],{"__TSPROSE_proseElement":215,"schema":8378,"children":8379},{"name":236,"type":218,"linkable":219,"__TSPROSE_schema":215},[8380,8387],{"__TSPROSE_proseElement":215,"schema":8381,"children":8382,"id":8386},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8383],{"__TSPROSE_proseElement":215,"schema":8384,"data":8385},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Starting from the expanded forms of the cube of a sum and a difference, recover their bracketed cube forms:","paragraph-p9oW9p2W1",{"__TSPROSE_proseElement":215,"schema":8388,"data":8389,"storageKey":8391,"id":8392},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8390,"freeze":219},"a^3 + 3a^2b + 3ab^2 + b^3 \\overset{\\text{?}}{\\implies} (a+b)^3 \\\\ a^3 - 3a^2b + 3ab^2 - b^3 \\overset{\\text{?}}{\\implies} (a-b)^3","$$ a^3 + 3a^2b + 3ab^2 + b^3 \\overset{\\text{?}}{\\implies} (a+b)^3 \\\\ a^3 - 3a^2b + 3ab^2 - b^3 \\overset{\\text{?}}{\\implies} (a-b)^3 $$","blockMath-7fiCltIQJ",{"__TSPROSE_proseElement":215,"schema":8394,"children":8395},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[8396],{"__TSPROSE_proseElement":215,"schema":8397,"children":8398,"id":8402},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8399],{"__TSPROSE_proseElement":215,"schema":8400,"data":8401},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"To solve this, you will need to factor common terms out of parts of the expression, split the results into smaller pieces, expand brackets again, and then factor common terms again.","paragraph-gM9n4crJm",{"__TSPROSE_proseElement":215,"schema":8404,"children":8405},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[8406,8427],{"__TSPROSE_proseElement":215,"schema":8407,"children":8408,"id":8426},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8409,8412,8416,8419,8423],{"__TSPROSE_proseElement":215,"schema":8410,"data":8411},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Factor out ",{"__TSPROSE_proseElement":215,"schema":8413,"data":8414,"storageKey":3773,"id":8415},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3772},"inlinerMath-eiXlIGTfO-4",{"__TSPROSE_proseElement":215,"schema":8417,"data":8418},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," from the first two terms, and ",{"__TSPROSE_proseElement":215,"schema":8420,"data":8421,"storageKey":3690,"id":8422},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3689},"inlinerMath-tBkcHvoEV-2",{"__TSPROSE_proseElement":215,"schema":8424,"data":8425},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," from the last two.","paragraph-ESJFSqF1A",{"__TSPROSE_proseElement":215,"schema":8428,"data":8429,"storageKey":8431,"id":8432},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8430,"freeze":219},"a^3 + 3a^2b + 3ab^2 + b^3 = a^2(a+3b) + b^2(3a+b) \\\\ a^3 - 3a^2b + 3ab^2 - b^3 = a^2(a-3b) + b^2(3a-b)","$$ a^3 + 3a^2b + 3ab^2 + b^3 = a^2(a+3b) + b^2(3a+b) \\\\ a^3 - 3a^2b + 3ab^2 - b^3 = a^2(a-3b) + b^2(3a-b) $$","blockMath-bfQIfRGGj",{"__TSPROSE_proseElement":215,"schema":8434,"children":8435},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[8436,8461,8467],{"__TSPROSE_proseElement":215,"schema":8437,"children":8438,"id":8460},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8439,8442,8448,8451,8457],{"__TSPROSE_proseElement":215,"schema":8440,"data":8441},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Split brackets of the form ",{"__TSPROSE_proseElement":215,"schema":8443,"data":8444,"storageKey":8446,"id":8447},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8445},"a \\pm 3b","$ a \\pm 3b $","inlinerMath-qItzIQZbe",{"__TSPROSE_proseElement":215,"schema":8449,"data":8450},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," so that ",{"__TSPROSE_proseElement":215,"schema":8452,"data":8453,"storageKey":8455,"id":8456},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8454},"a \\pm b","$ a \\pm b $","inlinerMath-Tz0eFZwYx",{"__TSPROSE_proseElement":215,"schema":8458,"data":8459},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," appears explicitly inside each of them:","paragraph-UCBBOGb7Y",{"__TSPROSE_proseElement":215,"schema":8462,"data":8463,"storageKey":8465,"id":8466},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8464,"freeze":219},"a^2(a+3b) + b^2(3a+b) = a^2((a+b) + 2b) + b^2((a+b) + 2a) \\\\ a^2(a-3b) + b^2(3a-b) = a^2((a-b) - 2b) + b^2((a-b) - 2a)","$$ a^2(a+3b) + b^2(3a+b) = a^2((a+b) + 2b) + b^2((a+b) + 2a) \\\\ a^2(a-3b) + b^2(3a-b) = a^2((a-b) - 2b) + b^2((a-b) - 2a) $$","blockMath-XpWmXZo89",{"__TSPROSE_proseElement":215,"schema":8468,"children":8469,"id":8481},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8470,8473,8479],{"__TSPROSE_proseElement":215,"schema":8471,"data":8472},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"After that, expand the brackets and factor out the common factor ",{"__TSPROSE_proseElement":215,"schema":8474,"data":8475,"storageKey":8477,"id":8478},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8476},"(a \\pm b)","$ (a \\pm b) $","inlinerMath-QO1tCOnny",{"__TSPROSE_proseElement":215,"schema":8480,"data":291},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-hjccX9EAF",{"__TSPROSE_proseElement":215,"schema":8483,"children":8484},{"name":3659,"type":218,"linkable":219,"__TSPROSE_schema":215},[8485,8504,8510,8523,8529,8536,8542,8563,8569,8582,8588,8595,8601],{"__TSPROSE_proseElement":215,"schema":8486,"children":8487,"id":8503},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8488,8491,8495,8497,8501],{"__TSPROSE_proseElement":215,"schema":8489,"data":8490},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"We will carry out the derivation for the cube-of-a-sum formula. The derivation for the cube of a difference is exactly the same. First factor out ",{"__TSPROSE_proseElement":215,"schema":8492,"data":8493,"storageKey":3773,"id":8494},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3772},"inlinerMath-eiXlIGTfO-5",{"__TSPROSE_proseElement":215,"schema":8496,"data":8418},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":8498,"data":8499,"storageKey":3690,"id":8500},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3689},"inlinerMath-tBkcHvoEV-3",{"__TSPROSE_proseElement":215,"schema":8502,"data":8425},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-ffS0N9seF",{"__TSPROSE_proseElement":215,"schema":8505,"data":8506,"storageKey":8508,"id":8509},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8507,"freeze":219},"a^2(a+3b) + b^2(3a+b)","$$ a^2(a+3b) + b^2(3a+b) $$","blockMath-RfAudvYCf",{"__TSPROSE_proseElement":215,"schema":8511,"children":8512,"id":8522},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8513,8516,8520],{"__TSPROSE_proseElement":215,"schema":8514,"data":8515},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Inside the brackets, form another pair of brackets of the form ",{"__TSPROSE_proseElement":215,"schema":8517,"data":8518,"storageKey":6003,"id":8519},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6002},"inlinerMath-mEpzc2nDI-1",{"__TSPROSE_proseElement":215,"schema":8521,"data":4028},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"paragraph-YmLp2EXLs",{"__TSPROSE_proseElement":215,"schema":8524,"data":8525,"storageKey":8527,"id":8528},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8526,"freeze":219},"a^2((a+b) + 2b) + b^2((a+b) + 2a)","$$ a^2((a+b) + 2b) + b^2((a+b) + 2a) $$","blockMath-MaNR8IH5M",{"__TSPROSE_proseElement":215,"schema":8530,"children":8531,"id":8535},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8532],{"__TSPROSE_proseElement":215,"schema":8533,"data":8534},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Expand both outer brackets:","paragraph-hPiw71Gir",{"__TSPROSE_proseElement":215,"schema":8537,"data":8538,"storageKey":8540,"id":8541},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8539,"freeze":219},"a^2(a+b) + 2a^2b + b^2(a+b) + 2ab^2","$$ a^2(a+b) + 2a^2b + b^2(a+b) + 2ab^2 $$","blockMath-b1Cy5nmsv",{"__TSPROSE_proseElement":215,"schema":8543,"children":8544,"id":8562},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8545,8548,8552,8555,8559],{"__TSPROSE_proseElement":215,"schema":8546,"data":8547},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Factor out the common factor ",{"__TSPROSE_proseElement":215,"schema":8549,"data":8550,"storageKey":6003,"id":8551},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6002},"inlinerMath-mEpzc2nDI-2",{"__TSPROSE_proseElement":215,"schema":8553,"data":8554},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," from the first and third terms, and the common factor ",{"__TSPROSE_proseElement":215,"schema":8556,"data":8557,"storageKey":3854,"id":8558},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":3853},"inlinerMath-McBVEYBhA-2",{"__TSPROSE_proseElement":215,"schema":8560,"data":8561},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," from the second and fourth terms:","paragraph-h9gCT5Keh",{"__TSPROSE_proseElement":215,"schema":8564,"data":8565,"storageKey":8567,"id":8568},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8566,"freeze":219},"(a+b)(a^2 + b^2) + 2ab(a+b)","$$ (a+b)(a^2 + b^2) + 2ab(a+b) $$","blockMath-K0JZmw77e",{"__TSPROSE_proseElement":215,"schema":8570,"children":8571,"id":8581},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8572,8574,8578],{"__TSPROSE_proseElement":215,"schema":8573,"data":8547},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":8575,"data":8576,"storageKey":6003,"id":8577},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6002},"inlinerMath-mEpzc2nDI-3",{"__TSPROSE_proseElement":215,"schema":8579,"data":8580},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," one more time:","paragraph-Z3Ykf1yKl",{"__TSPROSE_proseElement":215,"schema":8583,"data":8584,"storageKey":8586,"id":8587},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8585,"freeze":219},"(a+b)(a^2 + 2ab + b^2)","$$ (a+b)(a^2 + 2ab + b^2) $$","blockMath-BlukFRqWF",{"__TSPROSE_proseElement":215,"schema":8589,"children":8590,"id":8594},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8591],{"__TSPROSE_proseElement":215,"schema":8592,"data":8593},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Inside the second brackets, we got the expanded square-of-a-sum formula. 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Since we multiply by identical expressions, the value of the fraction does not change:","paragraph-G07dwibBG",{"__TSPROSE_proseElement":215,"schema":8704,"data":8705,"storageKey":8707,"id":8708},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8706,"freeze":215},"\\frac{4 \\yellow{(\\sqrt{5} - 1) (\\sqrt[4]{5} - 1) (\\sqrt[8]{5} - 1) (\\sqrt[16]{5} - 1) }}{(\\sqrt{5} + 1)\\yellow{(\\sqrt{5} - 1)}(\\sqrt[4]{5} + 1)\\yellow{(\\sqrt[4]{5} - 1)}(\\sqrt[8]{5} + 1)\\yellow{(\\sqrt[8]{5} - 1)}(\\sqrt[16]{5} + 1)\\yellow{(\\sqrt[16]{5} - 1)}}","$$ \\frac{4 \\yellow{(\\sqrt{5} - 1) (\\sqrt[4]{5} - 1) (\\sqrt[8]{5} - 1) (\\sqrt[16]{5} - 1) }}{(\\sqrt{5} + 1)\\yellow{(\\sqrt{5} - 1)}(\\sqrt[4]{5} + 1)\\yellow{(\\sqrt[4]{5} - 1)}(\\sqrt[8]{5} + 1)\\yellow{(\\sqrt[8]{5} - 1)}(\\sqrt[16]{5} + 1)\\yellow{(\\sqrt[16]{5} - 1)}} $$","blockMath-zJ0eDnI4j",{"__TSPROSE_proseElement":215,"schema":8710,"children":8711,"id":8721},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8712,8714,8718],{"__TSPROSE_proseElement":215,"schema":8713,"data":8672},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":8715,"data":8716,"storageKey":3960,"id":8717},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-26",{"__TSPROSE_proseElement":215,"schema":8719,"data":8720},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," formula to each pair of brackets, then carry out massive cancellation:","paragraph-CQ5JSg1dW",{"__TSPROSE_proseElement":215,"schema":8723,"data":8724,"storageKey":8726,"id":8727},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8725,"freeze":219},"\\frac{\\cancel{4} \\cancel{(\\sqrt{5} - 1)} \\cancel{(\\sqrt[4]{5} - 1)} \\cancel{(\\sqrt[8]{5} - 1)} (\\sqrt[16]{5} - 1)}{\\cancel{(5 - 1)} \\cancel{(\\sqrt{5} - 1)}\\cancel{(\\sqrt[4]{5} - 1)}\\cancel{(\\sqrt[8]{5} - 1)}} = \\\\ \\sqrt[16]{5} - 1","$$ \\frac{\\cancel{4} \\cancel{(\\sqrt{5} - 1)} \\cancel{(\\sqrt[4]{5} - 1)} \\cancel{(\\sqrt[8]{5} - 1)} (\\sqrt[16]{5} - 1)}{\\cancel{(5 - 1)} \\cancel{(\\sqrt{5} - 1)}\\cancel{(\\sqrt[4]{5} - 1)}\\cancel{(\\sqrt[8]{5} - 1)}} = \\\\ \\sqrt[16]{5} - 1 $$","blockMath-fnKkT32on",{"__TSPROSE_proseElement":215,"schema":8729,"children":8730,"id":8741},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8731,8734,8738],{"__TSPROSE_proseElement":215,"schema":8732,"data":8733},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"We have just simplified the expression equal to ",{"__TSPROSE_proseElement":215,"schema":8735,"data":8736,"storageKey":6055,"id":8737},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":6054},"inlinerMath-TLSwLL8iH-2",{"__TSPROSE_proseElement":215,"schema":8739,"data":8740},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},". Now substitute it into the final expression:","paragraph-RxaEXN2MN",{"__TSPROSE_proseElement":215,"schema":8743,"data":8744,"storageKey":8746,"id":8747},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8745,"freeze":219},"(1+x)^{48} = (\\cancel{1} + \\sqrt[16]{5} - \\cancel{1})^{48} = (\\sqrt[16]{5})^{48} = 5^{\\frac{48}{16}} = 5^3 = 125","$$ (1+x)^{48} = (\\cancel{1} + \\sqrt[16]{5} - \\cancel{1})^{48} = (\\sqrt[16]{5})^{48} = 5^{\\frac{48}{16}} = 5^3 = 125 $$","blockMath-PDma7ZCEH","tricky-radicals",{"__TSPROSE_proseElement":215,"schema":8750,"data":8751,"children":8755,"id":8913},{"name":4653,"type":218,"linkable":215,"__TSPROSE_schema":215},{"info":8752},{"title":8753,"level":8036,"attributes":8754},"Extra-Long 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them.","paragraph-TDmAXxY2j",{"__TSPROSE_proseElement":215,"schema":8803,"children":8804},{"name":265,"type":218,"linkable":219,"__TSPROSE_schema":215},[8805],{"__TSPROSE_proseElement":215,"schema":8806,"children":8807,"id":8818},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8808,8811,8815],{"__TSPROSE_proseElement":215,"schema":8809,"data":8810},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Using the ",{"__TSPROSE_proseElement":215,"schema":8812,"data":8813,"storageKey":3960,"id":8814},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-27",{"__TSPROSE_proseElement":215,"schema":8816,"data":8817},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," formula, find the square of the required number using the data you already 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original numbers, without squaring:","paragraph-xp6c5lmO3",{"__TSPROSE_proseElement":215,"schema":8842,"data":8843,"storageKey":8845,"id":8846},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8844,"freeze":219},"438271606 + 561728395 = 1000000001 \\\\ 561728395 - 438271606 = 123456789","$$ 438271606 + 561728395 = 1000000001 \\\\ 561728395 - 438271606 = 123456789 $$","blockMath-MCZhuA95A",{"__TSPROSE_proseElement":215,"schema":8848,"children":8849,"id":8853},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8850],{"__TSPROSE_proseElement":215,"schema":8851,"data":8852},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Multiplying these two numbers is actually very easy:","paragraph-ugPxKO9pY",{"__TSPROSE_proseElement":215,"schema":8855,"data":8856,"storageKey":8858,"id":8859},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8857,"freeze":219},"(10000000000 + 1) \\cdot 123456789 = 123456789 \\cdot 1000000000 + 123456789 = 123456789000000000 + 123456789 = 123456789123456789","$$ (10000000000 + 1) \\cdot 123456789 = 123456789 \\cdot 1000000000 + 123456789 = 123456789000000000 + 123456789 = 123456789123456789 $$","blockMath-KeJbea51E",{"__TSPROSE_proseElement":215,"schema":8861,"children":8862,"id":8873},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8863,8866,8870],{"__TSPROSE_proseElement":215,"schema":8864,"data":8865},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"By the ",{"__TSPROSE_proseElement":215,"schema":8867,"data":8868,"storageKey":3960,"id":8869},{"name":275,"type":245,"linkable":215,"__TSPROSE_schema":215},{"label":1342},"dependencyInliner-zJAa5vIur-28",{"__TSPROSE_proseElement":215,"schema":8871,"data":8872},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},", we get the following situation:","paragraph-I0xpGWjZk",{"__TSPROSE_proseElement":215,"schema":8875,"data":8876,"storageKey":8878,"id":8879},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8877,"freeze":219},"123456789123456789 = 561728395^2 - 438271606^2","$$ 123456789123456789 = 561728395^2 - 438271606^2 $$","blockMath-qeXZO4FTi",{"__TSPROSE_proseElement":215,"schema":8881,"children":8882,"id":8893},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8883,8886,8890],{"__TSPROSE_proseElement":215,"schema":8884,"data":8885},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"Now it is easy to find ",{"__TSPROSE_proseElement":215,"schema":8887,"data":8888,"storageKey":8781,"id":8889},{"name":3669,"type":245,"linkable":215,"__TSPROSE_schema":215},{"katex":8780},"inlinerMath-9lqCBTu25-1",{"__TSPROSE_proseElement":215,"schema":8891,"data":8892},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215}," itself:","paragraph-TdU6vl6St",{"__TSPROSE_proseElement":215,"schema":8895,"data":8896,"storageKey":8898,"id":8899},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8897,"freeze":219},"561728395^2 = 123456789123456789 + 438271606^2 = 123456789123456789 + 192082000625819236 = 315538789749275025","$$ 561728395^2 = 123456789123456789 + 438271606^2 = 123456789123456789 + 192082000625819236 = 315538789749275025 $$","blockMath-nzQSRTeqp",{"__TSPROSE_proseElement":215,"schema":8901,"children":8902,"id":8906},{"name":240,"type":218,"linkable":215,"__TSPROSE_schema":215},[8903],{"__TSPROSE_proseElement":215,"schema":8904,"data":8905},{"name":244,"type":245,"linkable":219,"__TSPROSE_schema":215},"All that remains is to find the sum of the digits of the resulting number:","paragraph-Jp44rXCOv",{"__TSPROSE_proseElement":215,"schema":8908,"data":8909,"storageKey":8911,"id":8912},{"name":250,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":8910,"freeze":219},"3 + 1 + 5 + 5 + 3 + 8 + 7 + 8 + 9 + 7 + 4 + 9 + 2 + 7 + 5 + 0 + 2 + 5 = \\boxed{91}","$$ 3 + 1 + 5 + 5 + 3 + 8 + 7 + 8 + 9 + 7 + 4 + 9 + 2 + 7 + 5 + 0 + 2 + 5 = \\boxed{91} $$","blockMath-B1CYTnagT","extra-long-numbers",{"$$ x^2 + 2xy + y^2 $$":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareSumFactorExamples":8916,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareDiffExamples":8923,"$$ (x + y)^2 $$":8915,"$$ (x - 2)^2 $$":8915,"$$ x^2 - 4x + 4 $$":8915,"$$ a^2 + 2a + 1 $$":8915,"$$ (a + 1)^2 $$":8915,"$$ (a + 11)^2 $$":8915,"$$ a^2 + 22a + 121 $$":8915,"$$ n^2 + m^2 + 2mn $$":8915,"$$ (n + m)^2 $$":8915,"$$ \\left(\\frac{1}{5} - m\\right)^2 $$":8915,"$$ \\frac{1}{25} - \\frac{2m}{5} + m^2 $$":8915,"$$ x^2 - 2xy + y^2 $$":8915,"$$ (x - y)^2 $$":8915,"$$ \\left(x + \\frac{2}{3}\\right)^2 $$":8915,"$$ x^2 + \\frac{4}{3}x + \\frac{4}{9} $$":8915,"$$ 81 - 18a + a^2 $$":8915,"$$ (9 - a)^2 $$":8915,"$$ (2x - 3)^2 $$":8915,"$$ 4x^2 - 12x + 9 $$":8915,"$$ a^2 - 12a + 36 $$":8915,"$$ (a - 6)^2 $$":8915,"$$ (3a + 5)^2 $$":8915,"$$ 9a^2 + 30a + 25 $$":8915,"$$ 4b^2 - 4b + 1 $$":8915,"$$ (2b - 1)^2 $$":8915,"$$ (-8 - 5b)^2 $$":8915,"$$ 64 + 80b + 25b^2 $$":8915,"$$ 1 + 10x + 25x^2 $$":8915,"$$ (1 + 5x)^2 $$":8915,"$$ (3x - 10y)^2 $$":8915,"$$ 9x^2 - 60xy + 100y^2 $$":8915,"$$ 100a^2 + 20a + 1 $$":8915,"$$ (10a + 1)^2 $$":8915,"$$ (11b - 2d)^2 $$":8915,"$$ 121b^2 - 44bd + 4d^2 $$":8915,"$$ 1 - 18y + 81y^2 $$":8915,"$$ (1 - 9y)^2 $$":8915,"$$ (6m + 5n)^2 $$":8915,"$$ 36m^2 + 60mn + 25n^2 $$":8915,"$$ 9 - 12x + 4x^2 $$":8915,"$$ (3 - 2x)^2 $$":8915,"$$ \\left(b + \\frac{5}{12}c\\right)^2 $$":8915,"$$ b^2 + \\frac{5}{6}bc + \\frac{25}{144}c^2 $$":8915,"$$ 9x^2 + 30x + 25 $$":8915,"$$ (3x + 5)^2 $$":8915,"$$ (4x - 9y)^2 $$":8915,"$$ 16x^2 - 72xy + 81y^2 $$":8915,"$$ 9a^2 - 30am + 25m^2 $$":8915,"$$ (3a - 5m)^2 $$":8915,"$$ (5y + 2z)^2 $$":8915,"$$ 25y^2 + 20yz + 4z^2 $$":8915,"$$ 16m^2 + 24mn + 9n^2 $$":8915,"$$ (4m + 3n)^2 $$":8915,"$$ \\left(-7y + \\frac{1}{7}x\\right)^2 $$":8915,"$$ 49y^2 - 2xy + \\frac{x^2}{49} $$":8915,"$$ 49 + 36x^2 + 84x $$":8915,"$$ (7 + 6x)^2 $$":8915,"$$ 4 + 49x^2 - 28x $$":8915,"$$ (2 - 7x)^2 $$":8915,"$$ 4x^2 + 36xy + 81y^2 $$":8915,"$$ (2x + 9y)^2 $$":8915,"$$ 1 - a^2 $$":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquaresExamples":8929,"$$ (1 + a)(1 - a) $$":8915,"$$ (8 - a)(8 + a) $$":8915,"$$ 64 - a^2 $$":8915,"$$ 16 - x^2 $$":8915,"$$ (4 + x)(4 - x) $$":8915,"$$ (b - 5)(b + 5) $$":8915,"$$ b^2 - 25 $$":8915,"$$ -y^2 + 81 $$":8915,"$$ (9 + y)(9 - y) $$":8915,"$$ (7 + b)(7 - b) $$":8915,"$$ 49 - b^2 $$":8915,"$$ \\frac{1}{9} - b^2 $$":8915,"$$ \\left(\\frac{1}{3} + b\\right)\\left(\\frac{1}{3} - b\\right) $$":8915,"$$ (y + 2)(2 - y) $$":8915,"$$ 4 - y^2 $$":8915,"$$ y^2 - \\frac{25}{36} $$":8915,"$$ \\left(y + \\frac{5}{6}\\right)\\left(y - \\frac{5}{6}\\right) $$":8915,"$$ (a - 4)(4 + a) $$":8915,"$$ a^2 - 16 $$":8915,"$$ 0.81 - x^2 $$":8915,"$$ (0.9 + x)(0.9 - x) $$":8915,"$$ (1 + c)(-1 + c) $$":8915,"$$ c^2 - 1 $$":8915,"$$ 16a^2 - 1 $$":8915,"$$ (4a + 1)(4a - 1) $$":8915,"$$ (6 - a)(-a - 6) $$":8915,"$$ a^2 - 36 $$":8915,"$$ 100 - 9y^2 $$":8915,"$$ (10 + 3y)(10 - 3y) $$":8915,"$$ (-b + 1)(b + 1) $$":8915,"$$ 1 - b^2 $$":8915,"$$ 36a^2 - 25b^2 $$":8915,"$$ (6a + 5b)(6a - 5b) $$":8915,"$$ (-x - 5)(x - 5) $$":8915,"$$ 25 - x^2 $$":8915,"$$ -9p^2 + 0.16q^2 $$":8915,"$$ (0.4q + 3p)(0.4q - 3p) $$":8915,"$$ (1 - 4x)(1 + 4x) $$":8915,"$$ 1 - 16x^2 $$":8915,"$$ \\frac{64}{9}n^2 - 4m^2 $$":8915,"$$ \\left(\\frac{8}{3}n + 2m\\right)\\left(\\frac{8}{3}n - 2m\\right) $$":8915,"$$ (5b - 7)(5b + 7) $$":8915,"$$ 25b^2 - 49 $$":8915,"$$ 0.04x^2 - 0.64y^2 $$":8915,"$$ (0.2x + 0.8y)(0.2x - 0.8y) $$":8915,"$$ (8x + 9)(9 - 8x) $$":8915,"$$ 81 - 64x^2 $$":8915,"$$ 9m^2n^2 - 1 $$":8915,"$$ (3mn + 1)(3mn - 1) $$":8915,"$$ (4y + 5b)(4y - 5b) $$":8915,"$$ 16y^2 - 25b^2 $$":8915,"$$ 81 - 16p^2q^2 $$":8915,"$$ (9 + 4pq)(9 - 4pq) $$":8915,"$$ (0.5m + 7x)(7x - 0.5m) $$":8915,"$$ 49x^2 - 0.25m^2 $$":8915,"$$ 0.01a^2b^2 - 100m^2 $$":8915,"$$ (0.1ab + 10m)(0.1ab - 10m) $$":8915,"$$ \\left(\\frac{9}{7}y - 0.2z\\right)\\left(\\frac{9}{7}y + 0.2z\\right) $$":8915,"$$ \\frac{81}{49}y^2 - 0.04z^2 $$":8915,"$$ 121a^2b^4 - 49c^2 $$":8915,"$$ (11ab^2 + 7c)(11ab^2 - 7c) $$":8915,"$$ \\left(0.8ab - \\frac{1}{3}c\\right)\\left(0.8ab + \\frac{1}{3}c\\right) $$":8915,"$$ 0.64a^2b^2 - \\frac{c^2}{9} $$":8915,"$$ -\\frac{9}{64}t^2 + 36k^4l^6 $$":8915,"$$ \\left(6k^2l^3 + \\frac{3}{8}t\\right)\\left(6k^2l^3 - \\frac{3}{8}t\\right) $$":8915,"$$ (-0.4mn + 1.1xy)(0.4mn + 1.1xy) $$":8915,"$$ 1.21x^2y^2 - 0.16m^2n^2 $$":8915,"$$ \\left(-\\frac{1}{5}pq - \\frac{2}{9}q\\right)\\left(\\frac{1}{5}pq - \\frac{2}{9}q\\right) $$":8915,"$$ \\frac{4q^2}{81} - \\frac{p^2q^2}{25} $$":8915,"$$ (m + n)^3 $$":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSumDiffExamples":8935,"$$ m^3 + 3m^2n + 3mn^2 + n^3 $$":8915,"$$ (c - d)^3 $$":8915,"$$ c^3 - 3c^2d + 3cd^2 - d^3 $$":8915,"$$ a^3 + 3a^2b + 3ab^2 + b^3 $$":8915,"$$ (a + b)^3 $$":8915,"$$ (2 + b)^3 $$":8915,"$$ 8 + 12b + 6b^2 + b^3 $$":8915,"$$ (3 - a)^3 $$":8915,"$$ 27 - 27a + 9a^2 - a^3 $$":8915,"$$ x^3 - 3x^2y + 3xy^2 - y^3 $$":8915,"$$ (x - y)^3 $$":8915,"$$ (x - 2)^3 $$":8915,"$$ x^3 - 6x^2 + 12x - 8 $$":8915,"$$ (a + 2b)^3 $$":8915,"$$ a^3 + 6a^2b + 12ab^2 + 8b^3 $$":8915,"$$ 125m^3 + 75m^2 + 15m + 1 $$":8915,"$$ (5m + 1)^3 $$":8915,"$$ (c - 3d)^3 $$":8915,"$$ c^3 - 9c^2d + 27cd^2 - 27d^3 $$":8915,"$$ \\left(4m + \\frac{1}{3}k\\right)^3 $$":8915,"$$ 64m^3 + 16m^2k + \\frac{4mk^2}{3} + \\frac{k^3}{27} $$":8915,"$$ 64 - 96a + 48a^2 - 8a^3 $$":8915,"$$ (4 - 2a)^3 $$":8915,"$$ \\left(\\frac{2}{3}a - 3b\\right)^3 $$":8915,"$$ \\frac{8a^3}{27} - 4a^2b + 18ab^2 - 27b^3 $$":8915,"$$ (a^2 + x^2)^3 $$":8915,"$$ a^6 + 3a^4x^2 + 3a^2x^4 + x^6 $$":8915,"$$ m^3 - 12m^2 + 48m - 64 $$":8915,"$$ (m - 4)^3 $$":8915,"$$ (2a^2 - 3b^2)^3 $$":8915,"$$ 8a^6 - 36a^4b^2 + 54a^2b^4 - 27b^6 $$":8915,"$$ (2x^3 - 3y^2)^3 $$":8915,"$$ 8x^9 - 36x^6y^2 + 54x^3y^4 - 27y^6 $$":8915,"$$ a^3 + 18a^2 + 108a + 216 $$":8915,"$$ (a + 6)^3 $$":8915,"$$ (10a^4 - 6b^2)^3 $$":8915,"$$ 1000a^{12} - 1800a^8b^2 + 1080a^4b^4 - 216b^6 $$":8915,"$$ \\left(\\frac{1}{2}a - \\frac{1}{3}b\\right)^3 $$":8915,"$$ \\frac{a^3}{8} - \\frac{a^2b}{4} + \\frac{ab^2}{6} - \\frac{b^3}{27} $$":8915,"$$ \\frac{a^3}{8} + \\frac{3a^2}{2} + 6a + 8 $$":8915,"$$ \\left(\\frac{a}{2} + 2\\right)^3 $$":8915,"$$ (0.5m + 0.1n)^3 $$":8915,"$$ 0.125m^3 + 0.075m^2n + 0.015mn^2 + 0.001n^3 $$":8915,"$$ \\frac{x^3}{27} - \\frac{x^2}{3} + x - 1 $$":8915,"$$ \\left(\\frac{x}{3} - 1\\right)^3 $$":8915,"$$ (2x - 3xy)^3 $$":8915,"$$ 8x^3 - 36x^3y + 54x^3y^2 - 27x^3y^3 $$":8915,"$$ \\frac{p^3}{8} + \\frac{9p^2q}{8} + \\frac{27pq^2}{8} + \\frac{27q^3}{8} $$":8915,"$$ \\left(\\frac{p}{2} + \\frac{3q}{2}\\right)^3 $$":8915,"$$ \\left(2a + \\frac{1}{2}ab\\right)^3 $$":8915,"$$ 8a^3 + 6a^3b + \\frac{3a^3b^2}{2} + \\frac{a^3b^3}{8} $$":8915,"$$ 8x^3 + 6x^2y + \\frac{3xy^2}{2} + \\frac{y^3}{8} $$":8915,"$$ \\left(2x + \\frac{y}{2}\\right)^3 $$":8915,"$$ \\left(0.1n^4 - \\frac{1}{2}n^3\\right)^3 $$":8915,"$$ 0.001n^{12} - 0.015n^{11} + 0.075n^{10} - 0.125n^9 $$":8915,"$$ \\frac{27a^3}{8} - \\frac{9a^2b}{2} + 2ab^2 - \\frac{8b^3}{27} $$":8915,"$$ \\left(\\frac{3a}{2} - \\frac{2b}{3}\\right)^3 $$":8915,"$$ (0.2x^2 - 0.3x^3)^3 $$":8915,"$$ 0.008x^6 - 0.036x^7 + 0.054x^8 - 0.027x^9 $$":8915,"$$ \\frac{a^3}{64} - \\frac{3a^2b}{32} + \\frac{3ab^2}{16} - \\frac{b^3}{8} $$":8915,"$$ \\left(\\frac{a}{4} - \\frac{b}{2}\\right)^3 $$":8915,"$$ 27 - 9a + a^2 - \\frac{a^3}{27} $$":8915,"$$ \\left(3 - \\frac{a}{3}\\right)^3 $$":8915,"$$ \\frac{x^3}{8} + \\frac{x^2y}{4} + \\frac{xy^2}{6} + \\frac{y^3}{27} $$":8915,"$$ \\left(\\frac{x}{2} + \\frac{y}{3}\\right)^3 $$":8915,"$$ (3x^2 + \\text{?})^2 = \\text{?} + \\text{?} + 16y^4 $$":8915,"$$ (3x^2 + 4y^2)^2 = 9x^4 + 24x^2y^2 + 16y^4 $$":8915,"$ 16y^4 $":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareSum":8941,"$ b^2 $":8915,"$$ 16y^4 = (4y^2)^2 $$":8915,"$ 4y^2 $":8915,"$$ (3x^2 + 4y^2)^2 = (3x^2)^2 + 2 \\cdot (3x^2) \\cdot (4y^2) + (4y^2)^2 = 9x^4 + 24x^2y^2 + 16y^4 $$":8915,"$$ (\\text{?} - 5b^3)^2 = 49a^6 - \\text{?} + \\text{?} $$":8915,"$$ (7a^3 - 5b^3)^2 = 49a^6 - 70a^3b^3 + 25b^6 $$":8915,"$ a^2 $":8915,"$$ 49a^6 = (7a^3)^2 $$":8915,"$ 7a^3 $":8915,"$$ (7a^3 - 5b^3)^2 = (7a^3)^2 - 2 \\cdot (7a^3) \\cdot (5b^3) + (5b^3)^2 = 49a^6 - 70a^3b^3 + 25b^6 $$":8915,"$$ (4a^3 + \\text{?})^2 = \\text{?} + 40a^3b^4 + \\text{?} $$":8915,"$$ (4a^3 + 5b^4)^2 = 16a^6 + 40a^3b^4 + 25b^8 $$":8915,"$ 2ab $":8915,"$ 40a^3b^4 $":8915,"$ b $":8915,"$$ 40a^3b^4 = 2 \\cdot 20a^3b^4 = 2 \\cdot \\underset{a}{(4a^3)} \\cdot \\underset{b}{(5b^4)} $$":8915,"$ 5b^4 $":8915,"$$ (4a^3 + 5b^4)^2 = (4a^3)^2 + 2 \\cdot (4a^3) \\cdot (5b^4) + (5b^4)^2 = 16a^6 + 40a^3b^4 + 25b^8 $$":8915,"$$ (\\text{?} - 12a)(\\text{?} + \\text{?}) = 9b^2 - \\text{?} $$":8915,"$$ (3b - 12a)(3b + 12a) = 9b^2 - 144a^2 $$":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquares":8944,"$ a^2 - b^2 $":8915,"$ 9b^2 $":8915,"$$ 9b^2 = (3b)^2 $$":8915,"$ 3b $":8915,"$ (3b + 12a) $":8915,"$ b^2 = (12a)^2 = 144a^2 $":8915,"$$ (3b - 12a)(3b + 12a) = (3b)^2 - (12a)^2 = 9b^2 - 144a^2 $$":8915,"$$ (\\text{?} - 5c)(\\text{?} + 5c) = 16d^2 - \\text{?} $$":8915,"$$ (4d - 5c)(4d + 5c) = 16d^2 - 25c^2 $$":8915,"$ (\\text{?} + 5c) $":8915,"$ 5c $":8915,"$ a^2 = 16d^2 $":8915,"$$ 16d^2 = (4d)^2 $$":8915,"$ 4d $":8915,"$ (5c)^2 = 25c^2 $":8915,"$$ (4d - 5c)(4d + 5c) = (4d)^2 - (5c)^2 = 16d^2 - 25c^2 $$":8915,"$$ (0{,}7p + \\text{?})(\\text{?} - 0{,}7p) = \\frac{1}{9}m^8 - 0{,}49p^2 $$":8915,"$$ \\left(0{,}7p + \\frac{1}{3}m^4\\right)\\left(\\frac{1}{3}m^4 - 0{,}7p\\right) = \\frac{1}{9}m^8 - 0{,}49p^2 $$":8915,"$ (b + a)(a - b) = a^2 - b^2 $":8915,"$ 0{,}7p $":8915,"$ a $":8915,"$$ \\frac{1}{9}m^8 = \\left(\\frac{1}{3}m^4\\right)^2 $$":8915,"$ (0{,}7p)^2 = 0{,}49p^2 $":8915,"$ a = \\frac{1}{3}m^4 $":8915,"$$ \\left(0{,}7p + \\frac{1}{3}m^4\\right)\\left(\\frac{1}{3}m^4 - 0{,}7p\\right) = \\left(\\frac{1}{3}m^4\\right)^2 - (0{,}7p)^2 = \\frac{1}{9}m^8 - 0{,}49p^2 $$":8915,"$$ (3m^2 + \\text{?})(\\text{?} - \\text{?}) = 9m^4 - n^6 $$":8915,"$$ (3m^2 + n^3)(3m^2 - n^3) = 9m^4 - n^6 $$":8915,"$ 3m^2 $":8915,"$$ 9m^4 = (3m^2)^2 $$":8915,"$ n^6 = (n^3)^2 $":8915,"$ b = n^3 $":8915,"$$ (3m^2 + n^3)(3m^2 - n^3) = (3m^2)^2 - (n^3)^2 = 9m^4 - n^6 $$":8915,"$$ (5a^4 + \\text{?})^3 = \\text{?} + \\text{?} + \\text{?} + 8b^{12} $$":8915,"$$ (5a^4 + 2b^4)^3 = 125a^{12} + 150a^8b^4 + 60a^4b^8 + 8b^{12} $$":8915,"$ 8b^{12} $":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSum":8950,"$ b^3 $":8915,"$$ 8b^{12} = (2b^4)^3 $$":8915,"$ 2b^4 $":8915,"$$ (5a^4 + 2b^4)^3 = (5a^4)^3 + 3 \\cdot (5a^4)^2 \\cdot (2b^4) + 3 \\cdot (5a^4) \\cdot (2b^4)^2 + (2b^4)^3 = 125a^{12} + 150a^8b^4 + 60a^4b^8 + 8b^{12} $$":8915,"$$ (\\text{?} - 2x)^3 = z^6 - \\text{?} + \\text{?} - \\text{?} $$":8915,"$$ (z^2 - 2x)^3 = z^6 - 6xz^4 + 12x^2z^2 - 8x^3 $$":8915,"$ a^3 $":8915,"$$ z^6 = (z^2)^3 $$":8915,"$ z^2 $":8915,"$$ (z^2 - 2x)^3 = (z^2)^3 - 3 \\cdot (z^2)^2 \\cdot (2x) + 3 \\cdot (z^2) \\cdot (2x)^2 - (2x)^3 = z^6 - 6xz^4 + 12x^2z^2 - 8x^3 $$":8915,"$$ (\\text{?} + 4)^3 = \\text{?} + \\text{?} + 240a^5 + \\text{?} $$":8915,"$$ (5a^5 + 4)^3 = 125a^{15} + 300a^{10} + 240a^5 + 64 $$":8915,"$ 3ab^2 $":8915,"$ b = 4 $":8915,"$ 240a^5 $":8915,"$$ 240a^5 = 3 \\cdot 80a^5 = 3 \\cdot \\underset{a}{(5a^5)} \\cdot \\underset{b^2}{4^2} $$":8915,"$ 5a^5 $":8915,"$$ (5a^5 + 4)^3 = (5a^5)^3 + 3 \\cdot (5a^5)^2 \\cdot 4 + 3 \\cdot (5a^5) \\cdot 4^2 + 4^3 = 125a^{15} + 300a^{10} + 240a^5 + 64 $$":8915,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-diff-of-squares":8956,"$ 91^2 - 90^2 $":8915,"$ 181 $":8915,"$$ 91^2 - 90^2 = (91 + 90)(91 - 90) = 181 \\cdot 1 = 181 $$":8915,"$ 1 $":8915,"$$ (a+b)^2 = a^2 + 2ab + b^2 \\\\ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$":8915,"$ (a-b)^2 $":8915,"$ (a-b)^3 $":8915,"$$ (a-b)^2 = (a + (-b))^2 >>{big} (a-b)^3 = (a + (-b))^3 $$":8915,"$$ (a + (-b))^2 = a^2 + 2a(-b) + (-b)^2 = \\boxed{a^2 - 2ab + b^2} \\\\ (a + (-b))^3 = a^3 + 3a^2(-b) + 3a(-b)^2 + (-b)^3 = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3} $$":8915,"$ \\blacksquare $":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareDiff":8958,"$$ (-a+b)^2 = \\text{?} $$":8915,"$ -a $":8915,"$$ ((-a) + b)^2 = (-a)^2 + 2 \\cdot (-a) \\cdot b + b^2 = \\boxed{a^2 - 2ab + b^2} $$":8915,"$ -2ab $":8915,"$$ (-a - b)^2 = \\text{?} $$":8915,"$ -b $":8915,"$$ ((-a) + (-b))^2 = (-a)^2 + 2 \\cdot (-a) \\cdot (-b) + (-b)^2 = \\boxed{a^2 + 2ab + b^2} $$":8915,"$$ (-a - b)^2 = (-(a + b))^2 = (-1)^2(a + b)^2 = a^2 + 2ab + b^2 $$":8915,"$ n $":8915,"$ n+1 $":8915,"$$ (n+1)^2 - n^2 = (\\cancel{n} + 1 - \\cancel{n})(n + 1 + n) = 1 \\cdot ((n+1) + n) = (n+1) + n $$":8915,"$ 4 $":8915,"$ 2n $":8915,"$ 2n + 2 $":8915,"$ 2 $":8915,"$$ (2n + 2)^2 - (2n)^2 = (\\cancel{2n} + 2 - \\cancel{2n})(2n + 2 + 2n) = 2 \\cdot (4n + 2) = 4 \\cdot (2n + 1) $$":8915,"$ n^2 + (n+1)^2 $":8915,"$ 2n(n+1) $":8915,"$$ n^2 + (n+1)^2 - 2n(n+1) $$":8915,"$$ n^2 - 2n(n+1) + (n+1)^2 = (n - (n+1))^2 = (-1)^2 = 1 $$":8915,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-product":8961,"$ 201 \\cdot 199 $":8915,"$ 39999 $":8915,"$ 201 = 200 + 1 $":8915,"$ 199 = 200 - 1 $":8915,"$$ 201 \\cdot 199 = (200 + 1)(200 - 1) = 200^2 - 1^2 = 40000 - 1 = 39999 $$":8915,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-difference":8963,"$ 43440^2 - 43438 \\cdot 43442 $":8915,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fpractice\u002F$simpleProduct":8965,"$ 43438 = 43440 - 2 $":8915,"$ 43442 = 43440 + 2 $":8915,"$ 43440 $":8915,"$$ 43438 \\cdot 43442 = (43440 - 2)(43440 + 2) = 43440^2 - 2^2 $$":8915,"$$ 43440^2 - 43438 \\cdot 43442 = 43440^2 - \\left(43440^2 - 2^2\\right) = \\cancel{43440^2} - \\cancel{43440^2} + 2^2 = 2^2 = 4 $$":8915,"$ 43440^2 $":8915,"$ 2^2 = 4 $":8915,"$$ (\\sqrt5 + \\sqrt6 + \\sqrt7)(\\sqrt5 + \\sqrt6 - \\sqrt7)(\\sqrt5 - \\sqrt6 + \\sqrt7)(-\\sqrt5 + \\sqrt6 + \\sqrt7) $$":8915,"$ 104 $":8915,"$$ ((\\sqrt5 + \\sqrt6) + \\sqrt7)((\\sqrt5 + \\sqrt6) - \\sqrt7) = \\\\ (\\sqrt5 + \\sqrt6)^2 - (\\sqrt7)^2 = \\\\ (5 + 2 \\cdot \\sqrt{30} + 6) - 7 = \\\\ 4 + 2 \\cdot \\sqrt{30} $$":8915,"$$ (\\sqrt5 - \\sqrt6 + \\sqrt7)(-\\sqrt5 + \\sqrt6 + \\sqrt7) = \\\\ (\\sqrt7 + (\\sqrt5 - \\sqrt6))(\\sqrt7 - (\\sqrt5 - \\sqrt6)) = \\\\ (\\sqrt7)^2 - (\\sqrt5 - \\sqrt6)^2 = \\\\ 7 - (5 - 2 \\cdot \\sqrt{30} + 6) = \\\\ -4 + 2 \\cdot \\sqrt{30} $$":8915,"$$ (4 + 2 \\cdot \\sqrt{30})(-4 + 2 \\cdot \\sqrt{30}) = \\\\ (2 \\cdot \\sqrt{30} + 4)(2 \\cdot \\sqrt{30} - 4) = \\\\ (2 \\cdot \\sqrt{30})^2 - 4^2 = \\\\ 4 \\cdot 30 - 16 = \\\\ 104 $$":8915,"$$ \\frac{4x^2 + 12x + 9}{4x^2 - 9} $$":8915,"$$ \\frac{2x + 3}{2x - 3} $$":8915,"$$ 4x^2 + 12x + 9 = (2x)^2 + 2 \\cdot (2x) \\cdot 3 + 3^2 = (2x + 3)^2 $$":8915,"$$ 4x^2 - 9 = (2x)^2 - 3^2 = (2x + 3)(2x - 3) $$":8915,"$ (2x+3) $":8915,"$$ \\frac{(2x + 3)^2}{(2x + 3)(2x - 3)} = \\frac{\\cancel{(2x + 3)}(2x + 3)}{\\cancel{(2x + 3)}(2x - 3)} = \\frac{2x + 3}{2x - 3} $$":8915,"$$ (x^2 + y^2)^2 - (x^2 - y^2)^2 $$":8915,"$$ 4x^2y^2 $$":8915,"$ x^2 + y^2 $":8915,"$ x^2 - y^2 $":8915,"$$ (x^2 + y^2)^2 - (x^2 - y^2)^2 = \\bigl((x^2+y^2) + (x^2-y^2)\\bigr)\\bigl((x^2+y^2) - (x^2-y^2)\\bigr) $$":8915,"$$ (x^2 + \\cancel{y^2} + x^2 - \\cancel{y^2})(\\cancel{x^2} + y^2 - \\cancel{x^2} + y^2) = 2x^2 \\cdot 2y^2 = 4x^2y^2 $$":8915,"$$ \\frac{a^2 - b^2}{a^2 + 2ab + b^2} $$":8915,"$$ \\frac{a - b}{a + b} $$":8915,"$$ \\frac{a^2 - b^2}{a^2 + 2ab + b^2} = \\frac{(a - b)(a + b)}{(a + b)^2} $$":8915,"$ (a+b) $":8915,"$$ \\frac{(a - b)\\cancel{(a + b)}}{\\cancel{(a + b)}(a + b)} = \\frac{a - b}{a + b} $$":8915,"$$ \\frac{x - \\dfrac{9}{x}}{1 - \\dfrac{3}{x}} $$":8915,"$$ x + 3 $$":8915,"$ x $":8915,"$$ \\frac{\\left(x - \\dfrac{9}{x}\\right) \\cdot x}{\\left(1 - \\dfrac{3}{x}\\right) \\cdot x} = \\frac{x^2 - 9}{x - 3} $$":8915,"$$ \\frac{x^2 - 9}{x - 3} = \\frac{(x + 3)\\cancel{(x - 3)}}{\\cancel{x - 3}} = x + 3 $$":8915,"$$ \\frac{a^2 + 2ab + b^2 - 9}{a + b + 3} $$":8915,"$$ a + b - 3 $$":8915,"$$ a^2 + 2ab + b^2 - 9 = (a + b)^2 - 9 $$":8915,"$$ (a + b)^2 - 9 = (a + b)^2 - 3^2 = (a + b + 3)(a + b - 3) $$":8915,"$ (a + b + 3) $":8915,"$$ \\frac{(a + b + 3)(a + b - 3)}{a + b + 3} = a + b - 3 $$":8915,"$$ (a^2 - b^2)^2 + 4a^2b^2 $$":8915,"$$ (a^2 + b^2)^2 $$":8915,"$ (a^2 - b^2)^2 $":8915,"$$ (a^2 - b^2)^2 = a^4 - 2a^2b^2 + b^4 $$":8915,"$$ a^4 - 2a^2b^2 + b^4 + 4a^2b^2 = a^4 + 2a^2b^2 + b^4 $$":8915,"$$ a^4 + 2a^2b^2 + b^4 = (a^2)^2 + 2 \\cdot a^2 \\cdot b^2 + (b^2)^2 = (a^2 + b^2)^2 $$":8915,"$$ \\frac{\\dfrac{a+b}{a-b} + \\dfrac{a-b}{a+b}}{\\dfrac{(a+b)^2 + (a-b)^2}{a^2 - b^2}} $$":8915,"$$ 1 $$":8915,"$ (a-b)(a+b) $":8915,"$$ \\frac{a+b}{a-b} + \\frac{a-b}{a+b} = \\frac{(a+b)^2 + (a-b)^2}{(a-b)(a+b)} $$":8915,"$$ (a+b)^2 + (a-b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2 $$":8915,"$$ \\frac{a+b}{a-b} + \\frac{a-b}{a+b} = \\frac{2a^2 + 2b^2}{a^2 - b^2} $$":8915,"$$ \\frac{\\dfrac{2a^2 + 2b^2}{a^2 - b^2}}{\\dfrac{(a+b)^2 + (a-b)^2}{a^2 - b^2}} = \\frac{2a^2 + 2b^2}{a^2 - b^2} \\cdot \\frac{a^2 - b^2}{(a+b)^2 + (a-b)^2} $$":8915,"$$ \\frac{\\cancel{2a^2 + 2b^2}}{\\cancel{a^2 - b^2}} \\cdot \\frac{\\cancel{a^2 - b^2}}{\\cancel{2a^2 + 2b^2}} = 1 $$":8915,"$$ 10^x = \\left( 10^{624} + 25 \\right)^2 - \\left( 10^{624} - 25 \\right)^2 $$":8915,"$ 10 $":8915,"$ 626 $":8915,"$$ \\left( 10^{624} + 25 \\right)^2 - \\left( 10^{624} - 25 \\right)^2 = \\\\ \\left( 10^{624} + \\cancel{25} + 10^{624} - \\cancel{25} \\right)\\left( \\cancel{10^{624}} + 25 - \\cancel{10^{624}} + 25 \\right) = \\\\ 2 \\cdot 10^{624} \\cdot 50 = 10^{624} \\cdot 100 = 10^{624} \\cdot 10^2 = 10^{626} $$":8915,"$$ 10^x = 10^{626} $$":8915,"$ x = 626 $":8915,"$$ \\left( 1 - \\frac{1}{2^2} \\right) \\left( 1 - \\frac{1}{3^2} \\right) \\left( 1 - \\frac{1}{4^2} \\right) \\cdots \\left( 1 - \\frac{1}{n^2} \\right) $$":8915,"$ n = 100 $":8915,"$$ \\frac{n+1}{2n} $$":8915,"$ n=100 $":8915,"$$ \\frac{101}{200} $$":8915,"$$ \\left( \\frac{2^2 - 1}{2^2} \\right) \\left( \\frac{3^2 - 1}{3^2} \\right) \\left( \\frac{4^2 - 1}{4^2} \\right) \\cdots \\left( \\frac{n^2 - 1}{n^2} \\right) $$":8915,"$$ \\left( \\frac{(2-1)(2+1)}{2^2} \\right) \\left( \\frac{(3-1)(3+1)}{3^2} \\right) \\left( \\frac{(4-1)(4+1)}{4^2} \\right) \\cdots \\left( \\frac{(n-1)(n+1)}{n^2} \\right) $$":8915,"$$ \\frac{1 \\cdot 3 \\cdot 2 \\cdot 4 \\cdot 3 \\cdot 5 \\cdots (n-1)(n+1)}{2^2 \\cdot 3^2 \\cdot 4^2 \\cdots n^2} $$":8915,"$$ \\frac{(\\cancel{1 \\cdot 2 \\cdot 3 \\cdots (n-1)})(\\cancel{3 \\cdot 4 \\cdot 5 \\cdots n} \\cdot (n+1))}{(2 \\cdot \\cancel{3 \\cdots (n-1)} \\cdot n) \\cdot (\\cancel{2 \\cdot 3 \\cdots n})} = \\frac{n+1}{2n} $$":8915,"$$ \\frac{n+1}{2n} = \\frac{100+1}{2 \\cdot 100} = \\frac{101}{200} $$":8915,"$$ 2^x - 2^y = 1 >>{big} 4^x - 4^y = \\frac{5}{3} >>{big} x - y = \\text{?} $$":8915,"$$ \\frac{a^{n}}{a^{m}} = a^{n-m} $$":8915,"$$ \\frac{5}{3} = 4^x - 4^y = (2^2)^x - (2^2)^y = 2^{2x} - 2^{2y} = (2^x)^2 - (2^y)^2 $$":8915,"$$ \\frac{5}{3} = (2^x)^2 - (2^y)^2 = (2^x + 2^y)\\underbrace{(2^x - 2^y)}_{\\small 1} \\\\ 2^x + 2^y = \\frac{5}{3} \\\\ $$":8915,"$$ (2^x + \\cancel{2^y}) + (2^x - \\cancel{2^y}) = \\frac{5}{3} + 1 \\\\ 2 \\cdot 2^x = \\frac{8}{3} \\\\ 2^x = \\frac{4}{3} $$":8915,"$$ (2^x + \\cancel{2^y}) - (2^x - \\cancel{2^y}) = \\frac{5}{3} - 1 \\\\ 2 \\cdot 2^y = \\frac{2}{3} \\\\ 2^y = \\frac{1}{3} $$":8915,"$$ \\frac{2^x}{2^y} = 2^{x-y} = \\frac{\\frac{4}{3}}{\\frac{1}{3}} = 4 = 2^2 \\\\ 2^{x-y} = 2^2 \\\\ $$":8915,"$ x-y $":8915,"$$ x - y = 2 $$":8915,"$ 693 $":8915,"$ m $":8915,"$ 10 \\cdot n + m $":8915,"$$ 59 = 10 \\cdot 5 + 9 >>{big} 13 = 10 \\cdot 1 + 3 >>{big} 92 = 10 \\cdot 9 + 2 >>{big} $$":8915,"$ 43 $":8915,"$ 34 $":8915,"$ 10n + m $":8915,"$ 10m + n $":8915,"$$ (10n + m)^2 - (10m + n)^2 = 693 \\\\ (10n + m + 10m + n)(10n + m - 10m - n) = 693 \\\\ (11n + 11m)(9n - 9m) = 693 \\\\ 99(n + m)(n - m) = 693 $$":8915,"$ 99 $":8915,"$$ (n + m)(n - m) = \\frac{693}{99} = 7 \\\\ (n + m)(n - m) = 7 $$":8915,"$ n+m $":8915,"$ n-m $":8915,"$ 7 $":8915,"$$ n + m = 7 \\\\ n - m = 1 $$":8915,"$ 9 $":8915,"$ n=4 $":8915,"$ m=3 $":8915,"$$ a^3 \\pm b^3 = (a \\pm b)(a^2 \\mp ab \\pm b^2) $$":8915,"$$ (a + 1)(a^2 - a + 1) $$":8915,"$$ a^3 + 1 $$":8915,"$$ a^3 + 8 $$":8915,"$$ (a + 2)(a^2 - 2a + 4) $$":8915,"$$ (x - 2)(x^2 + 2x + 4) $$":8915,"$$ x^3 - 8 $$":8915,"$$ x^3 - 27 $$":8915,"$$ (x - 3)(x^2 + 3x + 9) $$":8915,"$$ (2a + 3)(4a^2 - 6a + 9) $$":8915,"$$ 8a^3 + 27 $$":8915,"$$ 1 + y^3 $$":8915,"$$ (1 + y)(1 - y + y^2) $$":8915,"$$ (3 - b)(9 + 3b + b^2) $$":8915,"$$ 27 - b^3 $$":8915,"$$ m^3 - 27 $$":8915,"$$ (m - 3)(m^2 + 3m + 9) $$":8915,"$$ (1 + m^2)(1 - m^2 + m^4) $$":8915,"$$ 1 + m^6 $$":8915,"$$ p^3 - 1 $$":8915,"$$ (p - 1)(p^2 + p + 1) $$":8915,"$$ (3a - 4)(9a^2 + 12a + 16) $$":8915,"$$ 27a^3 - 64 $$":8915,"$$ 1 - 8x^3 $$":8915,"$$ (1 - 2x)(1 + 2x + 4x^2) $$":8915,"$$ (3x - 4)(9x^2 + 12x + 16) $$":8915,"$$ 27x^3 - 64 $$":8915,"$$ 27 - 8a^3 $$":8915,"$$ (3 - 2a)(9 + 6a + 4a^2) $$":8915,"$$ \\left(a + \\frac{1}{2}\\right)\\left(a^2 - \\frac{1}{2}a + \\frac{1}{4}\\right) $$":8915,"$$ a^3 + \\frac{1}{8} $$":8915,"$$ 8y^3 + x^3 $$":8915,"$$ (2y + x)(4y^2 - 2xy + x^2) $$":8915,"$$ \\left(\\frac{1}{2}x - \\frac{1}{3}y\\right)\\left(\\frac{1}{4}x^2 + \\frac{1}{6}xy + \\frac{1}{9}y^2\\right) $$":8915,"$$ \\frac{x^3}{8} - \\frac{y^3}{27} $$":8915,"$$ x^6 - y^9 $$":8915,"$$ (x^2 - y^3)(x^4 + x^2 y^3 + y^6) $$":8915,"$$ \\left(\\frac{1}{2}a - 2b\\right)\\left(\\frac{1}{4}a^2 + ab + 4b^2\\right) $$":8915,"$$ \\frac{a^3}{8} - 8b^3 $$":8915,"$$ 27a^3 - 8y^3 $$":8915,"$$ (3a - 2y)(9a^2 + 6ay + 4y^2) $$":8915,"$$ (16 + 4x + x^2)(x - 4) $$":8915,"$$ x^3 - 64 $$":8915,"$$ 1000x^{12}y^3 + 0{,}001a^9b^{15} $$":8915,"$$ (10x^4y + 0{,}1a^3b^5)(100x^8y^2 - x^4ya^3b^5 + 0{,}01a^6b^{10}) $$":8915,"$$ (49 + b^2 - 7b)(b + 7) $$":8915,"$$ b^3 + 343 $$":8915,"$ 111 $":8915,"$ 507 $":8915,"$ 2k + 1 $":8915,"$ k $":8915,"$$ 7 = 4^2 - 3^2 >>{big} 111 = 56^2 - 55^2 >>{big} 507 = 254^2 - 253^2 $$":8915,"$ 2k $":8915,"$$ 2k + 1 = \\left[ (k+1) + k \\right] = \\ldots $$":8915,"$$ \\ldots = \\left[ (k+1) + k \\right] \\cdot \\brand{1} = \\ldots $$":8915,"$$ \\ldots = \\left[ (k+1) + k \\right] \\cdot \\underbrace{\\left[ (k+1) - k \\right]}_{\\small 1} $$":8915,"$$ 2k + 1 = \\left[ (k+1) + k \\right] \\cdot \\left[ (k+1) - k \\right] = (k+1)^2 - k^2 \\\\ 2k + 1 = (k+1)^2 - k^2 $$":8915,"$$ 7 = 2 \\cdot 3 + 1 \\Rightarrow 3 >>{big} 111 = 2 \\cdot 55 + 1 \\Rightarrow 55 >>{big} 507 = 2 \\cdot 253 + 1 \\Rightarrow 253 $$":8915,"$$ 7 = \\boxed{4^2 - 3^2} >>{big} 111 = \\boxed{56^2 - 55^2} >>{big} 507 = \\boxed{254^2 - 253^2} $$":8915,"$$ a^3 + 3a^2b + 3ab^2 + b^3 \\overset{\\text{?}}{\\implies} (a+b)^3 \\\\ a^3 - 3a^2b + 3ab^2 - b^3 \\overset{\\text{?}}{\\implies} (a-b)^3 $$":8915,"$$ a^3 + 3a^2b + 3ab^2 + b^3 = a^2(a+3b) + b^2(3a+b) \\\\ a^3 - 3a^2b + 3ab^2 - b^3 = a^2(a-3b) + b^2(3a-b) $$":8915,"$ a \\pm 3b $":8915,"$ a \\pm b $":8915,"$$ a^2(a+3b) + b^2(3a+b) = a^2((a+b) + 2b) + b^2((a+b) + 2a) \\\\ a^2(a-3b) + b^2(3a-b) = a^2((a-b) - 2b) + b^2((a-b) - 2a) $$":8915,"$ (a \\pm b) $":8915,"$$ a^2(a+3b) + b^2(3a+b) $$":8915,"$$ a^2((a+b) + 2b) + b^2((a+b) + 2a) $$":8915,"$$ a^2(a+b) + 2a^2b + b^2(a+b) + 2ab^2 $$":8915,"$$ (a+b)(a^2 + b^2) + 2ab(a+b) $$":8915,"$$ (a+b)(a^2 + 2ab + b^2) $$":8915,"$$ (a+b)(a+b)^2 = (a+b)^3 $$":8915,"$$ x = \\frac{4}{(\\sqrt{5} + 1)(\\sqrt[4]{5} + 1)(\\sqrt[8]{5} + 1)(\\sqrt[16]{5} + 1)} $$":8915,"$ (1+x)^{48} $":8915,"$ 125 $":8915,"$$ \\frac{4 \\yellow{(\\sqrt{5} - 1) (\\sqrt[4]{5} - 1) (\\sqrt[8]{5} - 1) (\\sqrt[16]{5} - 1) }}{(\\sqrt{5} + 1)\\yellow{(\\sqrt{5} - 1)}(\\sqrt[4]{5} + 1)\\yellow{(\\sqrt[4]{5} - 1)}(\\sqrt[8]{5} + 1)\\yellow{(\\sqrt[8]{5} - 1)}(\\sqrt[16]{5} + 1)\\yellow{(\\sqrt[16]{5} - 1)}} $$":8915,"$$ \\frac{\\cancel{4} \\cancel{(\\sqrt{5} - 1)} \\cancel{(\\sqrt[4]{5} - 1)} \\cancel{(\\sqrt[8]{5} - 1)} (\\sqrt[16]{5} - 1)}{\\cancel{(5 - 1)} \\cancel{(\\sqrt{5} - 1)}\\cancel{(\\sqrt[4]{5} - 1)}\\cancel{(\\sqrt[8]{5} - 1)}} = \\\\ \\sqrt[16]{5} - 1 $$":8915,"$$ (1+x)^{48} = (\\cancel{1} + \\sqrt[16]{5} - \\cancel{1})^{48} = (\\sqrt[16]{5})^{48} = 5^{\\frac{48}{16}} = 5^3 = 125 $$":8915,"$$ 438271606^2 = 192082000625819236 $$":8915,"$ 561728395^2 $":8915,"$ 91 $":8915,"$$ 438271606 + 561728395 = 1000000001 \\\\ 561728395 - 438271606 = 123456789 $$":8915,"$$ (10000000000 + 1) \\cdot 123456789 = 123456789 \\cdot 1000000000 + 123456789 = 123456789000000000 + 123456789 = 123456789123456789 $$":8915,"$$ 123456789123456789 = 561728395^2 - 438271606^2 $$":8915,"$$ 561728395^2 = 123456789123456789 + 438271606^2 = 123456789123456789 + 192082000625819236 = 315538789749275025 $$":8915,"$$ 3 + 1 + 5 + 5 + 3 + 8 + 7 + 8 + 9 + 7 + 4 + 9 + 2 + 7 + 5 + 0 + 2 + 5 = \\boxed{91} $$":8915},null,{"type":8917,"schemaName":223,"elementTitle":8918,"content":8919,"resolvedHref":8920,"previewRequest":8921},"unique","Packing into the square of a sum",{"contentType":181,"contentTitle":160,"topicPart":212},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-sum-factor-examples",{"type":8917,"contentFullId":158,"contentType":181,"topicPart":212,"uniqueName":8922},"squareSumFactorExamples",{"type":8917,"schemaName":223,"elementTitle":8924,"content":8925,"resolvedHref":8926,"previewRequest":8927},"Examples Using the Square of a Difference",{"contentType":181,"contentTitle":160,"topicPart":212},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-diff-examples",{"type":8917,"contentFullId":158,"contentType":181,"topicPart":212,"uniqueName":8928},"squareDiffExamples",{"type":8917,"schemaName":223,"elementTitle":8930,"content":8931,"resolvedHref":8932,"previewRequest":8933},"Examples Using the Difference of 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Difference",{"contentType":181,"contentTitle":160,"topicPart":212},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum-diff-examples",{"type":8917,"contentFullId":158,"contentType":181,"topicPart":212,"uniqueName":8940},"cubeSumDiffExamples",{"type":8917,"schemaName":189,"elementTitle":187,"content":8942,"resolvedHref":188,"previewRequest":8943},{"contentType":181,"contentTitle":160,"topicPart":212},{"type":8917,"contentFullId":158,"contentType":181,"topicPart":212,"uniqueName":186},{"type":8917,"schemaName":189,"elementTitle":8945,"content":8946,"resolvedHref":8947,"previewRequest":8948},"Difference of squares 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formula",{"contentType":181,"contentTitle":160,"topicPart":212},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum",{"type":8917,"contentFullId":158,"contentType":181,"topicPart":212,"uniqueName":8955},"cubeSum",{"resolvedScriptSrc":8957},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-diff-of-squares.js",{"type":8917,"schemaName":189,"elementTitle":192,"content":8959,"resolvedHref":193,"previewRequest":8960},{"contentType":181,"contentTitle":160,"topicPart":212},{"type":8917,"contentFullId":158,"contentType":181,"topicPart":212,"uniqueName":191},{"resolvedScriptSrc":8962},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-product.js",{"resolvedScriptSrc":8964},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-difference.js",{"type":8917,"schemaName":4653,"elementTitle":5383,"content":8966,"resolvedHref":8967,"previewRequest":8968},{"contentType":181,"contentTitle":160,"topicPart":210},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=simple-product",{"type":8917,"contentFullId":158,"contentType":181,"topicPart":210,"uniqueName":8969},"simpleProduct",[8971,8978,8983,8988,8993,8999,9006,9011,9016,9022,9030,9038,9043,9048],{"link":8972,"schemaName":8973,"title":8974,"seo":8975,"description":8977},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=what-are-special-products","heading","What Are Special Products?",{"title":8976},"What are Special Products?","Special Products are formulas that let you quickly expand brackets or, going the other way, factor expressions back into brackets.",{"link":8972,"schemaName":8979,"title":160,"key":8980,"seo":8981,"description":8982},"accent_term",{"title":8976},{},"Formulas that let you quickly \"unpack\" compact powered expressions into expansions or, going the other way,\n          \"pack\" long sums into a compact form. These formulas save you from doing routine calculations by hand.",{"link":188,"schemaName":189,"title":8984,"key":8985,"seo":8986,"description":8987},"Square of a Sum",{},{},"One of the Special Products formulas: (a+b)² = a² + 2ab + b².\n          It lets you quickly expand brackets or, the other way around, pack an expanded expression back into them.",{"link":193,"schemaName":189,"title":8989,"key":8990,"seo":8991,"description":8992},"Square of a Difference",{},{},"One of the Special Products formulas: (a-b)² = a² - 2ab + b².\n          It lets you expand parentheses quickly or, going the other way, factor an expression back into squared parentheses.",{"link":8994,"schemaName":8973,"title":8995,"seo":8996,"description":8998},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=applications-of-the-square-and-a-difference","Applications of the Square and a Difference",{"title":8997},"Where are the square of a sum and the square of a difference used?","Practical examples of using the Special Products formulas for the square of a sum and the square of a difference in different areas of mathematics and real life.",{"link":9000,"schemaName":223,"title":9001,"key":9002,"seo":9003,"description":9005},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=fast-square","Quick Squares",{},{"title":9004},"How can you square a number quickly in your head?","A universal method for quickly squaring any number by using the square-of-a-sum and square-of-a-difference formulas.\n          For numbers within 100, this process can often be done mentally.",{"link":8947,"schemaName":189,"title":9007,"key":9008,"seo":9009,"description":9010},"Difference of Squares",{},{},"One of the Special Products formulas: a² - b² = (a + b)(a - b).\n          Lets you halve or double the degree.\n          Used to simplify expressions.",{"link":8953,"schemaName":189,"title":9012,"key":9013,"seo":9014,"description":9015},"Cube of a Sum",{},{},"One of the Special Products formulas: (a + b)³ = a³ + 3a²b + 3ab² + b³.\n          Lets you quickly expand the cube of a sum of two expressions.\n          Used to simplify expressions and solve equations.",{"link":9017,"schemaName":189,"title":9018,"key":9019,"seo":9020,"description":9021},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-diff","Cube of a Difference",{},{},"One of the Special Products formulas: (a - b)³ = a³ - 3a²b + 3ab² - b³.\n          Lets you quickly expand the cube of a difference of two expressions.\n          Used to simplify expressions and solve equations.",{"link":9023,"schemaName":9024,"title":9025,"key":9026,"seo":9027,"description":9029},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=memorization-tips","list","How to memorize?",{},{"title":9028},"How to memorize Special Products?","A set of tips and tricks that help you memorize Special Products and avoid mixing them up.",{"link":9031,"schemaName":9032,"title":9033,"key":9034,"seo":9036,"description":9037},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=special-products-table","table","Table of Special Products",{"title":9035},"Special Products table",{},"A single table with all the Special Products formulas.",{"link":9039,"schemaName":223,"title":9040,"seo":9041,"description":9042},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=practice-square-of-a-sum-and-square-of-a-difference","Problems on the square of a sum and the square of a difference",{},"A large set of problems for practicing the Special Products formulas for the square of a sum and the square of a difference.\n          Two types of problems: expand and rewrite as the square of a sum or a difference.",{"link":9044,"schemaName":223,"title":9045,"seo":9046,"description":9047},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=practice-difference-of-squares","Problems on the difference of squares",{},"A large set of problems for practicing the difference of squares formula.\n          Two types of problems: expand and rewrite as a difference of squares.",{"link":9049,"schemaName":223,"title":9050,"seo":9051,"description":9052},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=practice-cube-of-a-sum-and-cube-of-a-difference","Problems on the cube of a sum and the cube of a difference",{},"A large set of problems for practicing the Special Products formulas for the cube of a sum and the cube of a difference.\n          Two types of problems: expand and rewrite as the cube of a sum or a difference.",[9054,9056,9057,9058,9059,9060,9061,9062,9063,9064,9065,9066,9067,9068,9069,9070,9071,9072,9073,9074,9075],{"type":9055,"schemaName":223,"title":225,"elementId":1297},"element",{"type":9055,"schemaName":223,"title":1301,"elementId":2486},{"type":9055,"schemaName":223,"title":2490,"elementId":3615},{"type":9055,"schemaName":223,"title":3619,"elementId":4650},{"type":9055,"schemaName":4653,"title":4656,"elementId":4750},{"type":9055,"schemaName":4653,"title":4755,"elementId":4860},{"type":9055,"schemaName":223,"title":4864,"elementId":5057},{"type":9055,"schemaName":223,"title":5061,"elementId":5378},{"type":9055,"schemaName":4653,"title":5383,"elementId":5474},{"type":9055,"schemaName":4653,"title":5479,"elementId":5623},{"type":9055,"schemaName":4653,"title":5628,"elementId":5736},{"type":9055,"schemaName":223,"title":5740,"elementId":6418},{"type":9055,"schemaName":4653,"title":6423,"elementId":6536},{"type":9055,"schemaName":4653,"title":6541,"elementId":6754},{"type":9055,"schemaName":4653,"title":6759,"elementId":6921},{"type":9055,"schemaName":4653,"title":6926,"elementId":7228},{"type":9055,"schemaName":223,"title":7232,"elementId":8030},{"type":9055,"schemaName":4653,"title":8035,"elementId":8369},{"type":9055,"schemaName":4653,"title":8374,"elementId":8609},{"type":9055,"schemaName":4653,"title":8614,"elementId":8748},{"type":9055,"schemaName":4653,"title":8753,"elementId":8913},{"contributorsCount":173,"sponsorsCount":9077},0,{"shortBookId":5,"frontNav":9079},{"type":14,"shortId":5,"title":15,"flags":9080,"link":17,"children":9081},{},[9082,9091],{"type":9083,"separator":215,"shortId":9084,"title":166,"flags":9085,"link":167,"children":9086},"group","foundations\u002Fpolynomials",{},[9087],{"type":181,"shortId":158,"title":160,"flags":9088,"link":9089,"parts":9090},{},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F",[212,213,210],{"type":9083,"separator":215,"shortId":9092,"title":9093,"flags":9094,"link":9095,"children":9096},"foundations\u002Fequations","Equations",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002F",[9097,9103,9108,9165],{"type":181,"shortId":9098,"title":9099,"flags":9100,"link":9101,"parts":9102},"foundations\u002Fequations\u002Felementary","Elementary Equations",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Felementary\u002F",[212,213,210],{"type":8,"shortId":9104,"title":9105,"flags":9106,"link":9107},"foundations\u002Fequations\u002Fzero-product-property","Zero Product Property",{},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fzero-product-property\u002F",{"type":9083,"separator":219,"shortId":9109,"title":9110,"flags":9111,"link":9112,"children":9113},"foundations\u002Fequations\u002Fquadratic","Quadratic Equations",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002Fquadratic\u002F",[9114,9120,9126,9130,9136,9142,9147,9153,9159],{"type":181,"shortId":9115,"title":9116,"flags":9117,"link":9118,"parts":9119},"foundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it","What is it?",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it\u002F",[212,213,210],{"type":181,"shortId":9121,"title":9122,"flags":9123,"link":9124,"parts":9125},"foundations\u002Fequations\u002Fquadratic\u002Fincomplete","With missing terms",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete\u002F",[212,213,210],{"type":181,"shortId":9127,"title":182,"flags":9128,"link":183,"parts":9129},"foundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square",{},[212,213,210],{"type":181,"shortId":9131,"title":9132,"flags":9133,"link":9134,"parts":9135},"foundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula","Quadratic formula",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002F",[212,213,210],{"type":181,"shortId":9137,"title":9138,"flags":9139,"link":9140,"parts":9141},"foundations\u002Fequations\u002Fquadratic\u002Ffactoring","Factoring quadratics",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002F",[212,213,210],{"type":8,"shortId":9143,"title":9144,"flags":9145,"link":9146},"foundations\u002Fequations\u002Fquadratic\u002Freal-world","Real Life",{"secondary":215},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F",{"type":181,"shortId":9148,"title":9149,"flags":9150,"link":9151,"parts":9152},"foundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas","Vieta's Formulas",{"secondary":215},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002F",[212,213,210],{"type":181,"shortId":9154,"title":9155,"flags":9156,"link":9157,"parts":9158},"foundations\u002Fequations\u002Fquadratic\u002Fmental-solving","Mental solving",{"secondary":215,"advanced":215},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002F",[212,213],{"type":181,"shortId":9160,"title":9161,"flags":9162,"link":9163,"parts":9164},"foundations\u002Fequations\u002Fquadratic\u002Fformulas","General formulas",{"secondary":215},"\u002Fsummary\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fformulas\u002F",[213,210],{"type":181,"shortId":9166,"title":9167,"flags":9168,"link":9169,"parts":9170},"foundations\u002Fequations\u002Fequivalence","Equivalence of equations",{"dev":215},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fequivalence\u002F",[212],{"cameo":9172},[9173,9174,9175,9176,9177,9178,9179,9180,9181],"cameo-black-hole","cameo-confucius","cameo-dunno","cameo-einstein","cameo-euclid","cameo-gagarin","cameo-gauss","cameo-newton","cameo-t800",1777652715012]