Exponents & Roots

What are Exponents?

An exponent (also called a power or index) tells you how many times to multiply a base by itself.

2³ = 2 × 2 × 2 = 8

• Base: 2 (the number being multiplied)
• Exponent: 3 (how many times to multiply)
• Read as: "2 to the power of 3" or "2 cubed"

Exponents are a shorthand for repeated multiplication. Instead of writing 5 × 5 × 5 × 5, you write 5⁴.

Key Vocabulary

Term	Meaning	Example
Base	The number being raised to a power	5 in 5³
Exponent / Index	How many times to multiply	3 in 5³
Power	The result of the calculation	5³ = 125
Perfect square	A number that is a whole number squared	25 = 5²
Perfect cube	A number that is a whole number cubed	8 = 2³

Squares and Perfect Squares

Squaring means raising to the power of 2:

• 5² = 5 × 5 = 25
• 7² = 7 × 7 = 49

The first twelve perfect squares to memorise:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Cubes and Perfect Cubes

Cubing means raising to the power of 3:

• 3³ = 3 × 3 × 3 = 27
• 4³ = 4 × 4 × 4 = 64

Common perfect cubes: 1, 8, 27, 64, 125, 216

Square Roots

The square root (√) is the inverse of squaring. It asks: "What number, multiplied by itself, gives this result?"

• √25 = 5 (because 5² = 25)
• √144 = 12 (because 12² = 144)
• √0 = 0

Note: the square root of a negative number is not a real number (you cannot square a real number and get a negative result).

Cube Roots

The cube root (∛) is the inverse of cubing:

• ∛8 = 2 (because 2³ = 8)
• ∛27 = 3 (because 3³ = 27)
• ∛125 = 5 (because 5³ = 125)

Unlike square roots, cube roots of negative numbers do exist: ∛(-8) = -2.

Key Rules of Exponents

Rule	Formula	Example
Zero exponent	a⁰ = 1	7⁰ = 1
Power of one	a¹ = a	5¹ = 5
Multiply (same base)	aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷ = 128
Divide (same base)	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	3⁵ ÷ 3² = 3³ = 27
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(2³)² = 2⁶ = 64
Negative exponent	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8

Worked Examples

Example 1: Calculate 3⁴

• 3⁴ = 3 × 3 × 3 × 3 = 9 × 9 = 81

Example 2: Simplify 2³ × 2⁴

• Same base, so add exponents: 2³⁺⁴ = 2⁷ = 128

Example 3: Simplify (3²)³

• Power of a power: multiply exponents: 3²ˣ³ = 3⁶ = 729

Example 4: What is 5⁰?

• Any number to the power of zero = 1

Estimating Square Roots

For square roots that are not perfect, estimate by finding the two nearest perfect squares.

√50: since 7² = 49 and 8² = 64, we know √50 is just above 7. More precisely, √50 ≈ 7.07.

This skill is useful for checking calculator answers and for working with the Pythagorean theorem.

Common Mistakes

• Multiplying the base by the exponent: 2³ ≠ 6. It means 2 × 2 × 2 = 8.
• Adding instead of multiplying exponents when multiplying: 2³ × 2⁴ = 2⁷, not 2¹².
• Forgetting that a⁰ = 1 for any non-zero value of a.
• Thinking √(a + b) = √a + √b — this is false. √(9 + 16) = √25 = 5, not 3 + 4 = 7.

Tips and Tricks

• Memorise perfect squares up to 15² = 225 — they come up constantly in exams.
• For large powers of 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1024.
• To square a number ending in 5: multiply the tens digit by (tens digit + 1), then append 25. For example, 35² → 3 × 4 = 12, so 35² = 1225.
• The square root of a fraction: √(a/b) = √a / √b. For example, √(9/16) = 3/4.