Powers and roots

Foundation and Higher

The idea

A power tells you how many times to multiply a number by itself: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$ . A root undoes a power: $\sqrt[3]{64} = 4$ because $4^3 = 64$ .

Learn the squares to $15^2$ and the cubes of $1, 2, 3, 4, 5$ and $10$ .
Any non-zero number to the power $0$ is $1$ .
A positive number has two square roots: $\sqrt{49} = 7$ , and $x^2 = 49$ gives $x = \pm 7$ .

Watch out $3^4$ is $81$ , not $3 \times 4 = 12$ .

See it worked

Worked exampleNon-calculator1 mark

Work out the cube root of 216.

The cube root is the number whose cube is 216.
6 $\times$ 6 $\times$ 6 = 216, so the cube root is 6.

Answer $6$

Your turn

Question 1Non-calculator4 marks

Answer each part without a calculator.

(a) Work out $4^3$

(b) Write down the value of $\sqrt{81}$

(d) Write $2 \times 2 \times 2 \times 2 \times 2$ as a power of $2$ .

Question 2Non-calculator5 marks

A positive integer is $2^3\times3^2\times5$ .

(a) Find the smallest positive integer multiplier that makes it a square number.

(b) Find the square root of the resulting square number.

Question 3Calculator allowed5 marks

Find the smallest positive integer n for which n/18 is a square integer and n/30 is a cube integer. Show how prime factors determine your answer.

In past papers

235 official questions in 152 papers since 2017. The latest:

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