Powers and roots

A power tells you how many times to multiply a number by itself: 5³ = 5 × 5 × 5. A root goes backwards: √49 = 7 because 7² = 49. When you multiply powers of the same number you add the indices; when you divide you subtract them.

Key facts

  • a² means a × a; a³ means a × a × a.
  • √ undoes squaring; ∛ undoes cubing.
  • aᵐ × aⁿ = aᵐ⁺ⁿ
  • aᵐ ÷ aⁿ = aᵐ⁻ⁿ
  • Anything to the power 0 is 1.

Worked examples

Example 1

Work out 2⁵.

  1. Write it out. 2 × 2 × 2 × 2 × 2
  2. Multiply step by step: 2, 4, 8, 16, 32. 2⁵ = 32

Answer: 32

Example 2

Simplify 3⁴ × 3⁵ ÷ 3⁶. Give your answer as a power of 3, then as a number.

  1. Multiplying: add the powers. 3⁴ × 3⁵ = 3⁹
  2. Dividing: subtract the powers. 3⁹ ÷ 3⁶ = 3³
  3. Work it out. 3³ = 27

Answer: 3³ = 27

Example 3

A square floor tile has an area of 144 cm². How long is each side?

  1. Area of a square = side × side, so side = √area. √144
  2. 12 × 12 = 144. side = 12 cm

Answer: 12 cm