Practice

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Square of a sum and a difference, difference of squares, and cube of a sum and a difference — very useful formulas that let you quickly expand or factor expressions with powers.

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  • Term1
  • Statement5
  • Important2
  • Problem28
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Article
Summary
Practice

Practice: Square of a Sum and Square of a Difference

😀
Elementary
Write as the square of a sum or a difference:
x2+2xy+y2x^2 + 2xy + y^2

Practice: Difference of Squares

😀
Elementary
Write as the product of two binomials:
1−a21 - a^2

Practice: Cube of a Sum and Cube of a Difference

😀
Elementary
Expand:
(m+n)3(m + n)^3

Too Many Question Marks

😀
Elementary
Fill in the correct entries in place of the question marks.
(3x2+?)2=?+?+16y4(3x^2 + \text{?})^2 = \text{?} + \text{?} + 16y^4

Simple Difference of Squares

🤔
Intermediate
Without calculating any squares, find the value of 912−90291^2 - 90^2.

From One to the Other

😀
Elementary
Suppose you only know the formulas for the square and cube of a sum:
(a+b)2=a2+2ab+b2(a+b)3=a3+3a2b+3ab2+b3(a+b)^2 = a^2 + 2ab + b^2 \\ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Use them to derive the formulas for the square and cube of a difference (a−b)2(a-b)^2 and (a−b)3(a-b)^3.

Awkward Minus Signs

Method
😀
Elementary
Check whether the formula for the square of a difference still works when the negative term comes first rather than second:
(−a+b)2=?(-a+b)^2 = \text{?}

Prove This, Prove That…

😀
Elementary
Prove that the difference of the squares of two consecutive natural numbers equals the sum of those numbers.

Simple Product

🤔
Intermediate
Without multiplying directly, find the value of 201⋅199201 \cdot 199.

Simple Difference with a Product

🤔
Intermediate
Compute the expression 434402−43438⋅4344243440^2 - 43438 \cdot 43442.

Radical Mayhem

🤔
Intermediate
Find the value of the expression:
(5+6+7)(5+6−7)(5−6+7)(−5+6+7)(\sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 - \sqrt7)(\sqrt5 - \sqrt6 + \sqrt7)(-\sqrt5 + \sqrt6 + \sqrt7)

Time to Simplify

🤔
Intermediate
Simplify the expression:
4x2+12x+94x2−9\frac{4x^2 + 12x + 9}{4x^2 - 9}

Power of Ten

🤔
Intermediate
What number must be substituted for x to make the equality true?
10x=(10624+25)2−(10624−25)210^x = \left( 10^{624} + 25 \right)^2 - \left( 10^{624} - 25 \right)^2

Fraction Snake

Elegant
🤔
Intermediate
Simplify the expression:
(1−122)(1−132)(1−142)⋯(1−1n2)\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)
What does it equal when n = 100?

Difference of Exponents

🤔
Intermediate
2x−2y=12^x - 2^y = 1
4x−4y=53 4^x - 4^y = \frac{5}{3}
x−y=? x - y = \text{?}

The Mystery of 693

Elegant
🤔
Intermediate
The difference of squares of two two-digit numbers written with the same digits equals 693. Find those numbers.

Sum and Difference of Cubes

Method
🤔
Intermediate
Unlike squares, where there is a special difference-of-squares formula, cubes have two separate formulas: the sum of cubes and the difference of cubes.
a3±b3=(a±b)(a2∓ab±b2)a^3 \pm b^3 = (a \pm b)(a^2 \mp ab \pm b^2)
Expand:
(a+1)(a2−a+1)(a + 1)(a^2 - a + 1)

Any Odd is a Difference of Squares

Elegant
🤯
Advanced
Prove that any odd number can be written as a difference of squares of two numbers. Find the numbers whose difference of squares equals 7, 111, and 507.

Factoring the Cube of a Sum and a Difference

🤯
Advanced
Starting from the expanded forms of the cube of a sum and a difference, recover their bracketed cube forms:
a3+3a2b+3ab2+b3  ⟹  ?(a+b)3a3−3a2b+3ab2−b3  ⟹  ?(a−b)3a^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

Tricky Radicals

🤯
Advanced
It is known that
x=4(5+1)(54+1)(58+1)(516+1)x = \frac{4}{(\sqrt{5} + 1)(\sqrt[4]{5} + 1)(\sqrt[8]{5} + 1)(\sqrt[16]{5} + 1)}
What is the value of (1+x)48(1+x)^{48}?

Extra-Long Numbers

🤯
Advanced
It is known that
4382716062=192082000625819236438271606^2 = 192082000625819236
Without squaring by hand, find the sum of the digits of 5617283952561728395^2.
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