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MathidiaMathematics Academy
Integers, powers and roots
Lesson 3 of 75
Standard practice

Squares, cubes and roots

Objective. Recognise and use square numbers, cube numbers and their roots.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Squares, cubes and roots describe a number multiplied by itself twice or three times, and the inverse operations that undo this. The square of a number n is n² = n × n, such as 7² = 49, while the cube is n³ = n × n × n, such as 4³ = 64. The square root √n finds a value that squares to give n, and the cube root ∛n finds a value that cubes to give n, so √49 = 7 and ∛64 = 4. Learn the squares up to 15² and cubes up to 5³ by heart, since they appear constantly in exam questions.

Worked example

Find the value of 5³ − √36 using squares, cubes and roots.

Diagram of a square
  1. 1Calculate the cube: 5³ = 5 × 5 × 5 = 125
  2. 2Calculate the root: √36 = 6
  3. 3Subtract: 125 − 6
  4. 4Simplify: 125 − 6 = 119

Answer: 119

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Common mistake. Pupils sometimes think √49 means 49 ÷ 2, confusing square roots with halving; in fact √49 asks which number multiplied by itself gives 49, which is 7, not 24.5.

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