Maths

Powers and roots - Foundation

In this Revision Bite we are going to look at:

Squares, cubes and their roots

Squaring a number

32 means '3 squared', or 3 x 3.

The small 2 is an index number, or power. It tells us how many times we should multiply 3 by itself.

Similarly 72 means '7 squared', or 7 x 7.

And 102 means '10 squared', or 10 x 10.

So, 12 = 1 x 1 = 1

22 = 2 x 2 = 4

32 = 3 x 3 = 9

42 = 4 x 4 = 16

52 = 5 x 5 = 25

etc

1, 4, 9, 16, 25… are known as square numbers.

Square roots

The opposite of a square number is a square root.

We use the symbol square root of 4 = 2 to mean square root.

So we can say that square root of 4 = 2 and square root of 25 = 5.

However, this is not the whole story, because -2 x -2 is also 4, and -5 x -5 is also 25.

So, in fact, square root of 4 = 2 or -2. And square root of 25 = 5 or -5.

Remember that every positive number has two square roots.

Cubing a number

2 x 2 x 2 means '2 cubed', and is written as 23.

13 = 1 x 1 x 1 = 1

23 = 2 x 2 x 2 = 8

33 = 3 x 3 x 3 = 27

43 = 4 x 4 x 4 = 64

53 = 5 x 5 x 5 = 125

etc

1, 8, 27, 64, 125… are known as cube numbers.

Cube roots

The opposite of a cube number is a cube root. We use the symbol cube root to mean cube root.

So cube root of 8 is 2 and cube root of 27 is 3.

Each number only has one cube root.

Rules of indices

You can perform operations on numbers that have been squared cubed or raised to higher powers. There are three rules to remember for multiplying, dividing, and the power of a power.

Multiplying

When multiplying add the powers.

23 × 24 = (2 × 2 × 2) × (2 × 2 × 2 × 2)

= 27

Dividing

When dividing subtract the powers.

25 ÷ 22 = 2 x 2 x 2 x 2 x 2 over 2 x 2 = 2 × 2 × 2 (Cancelling two of the 2s)

= 23

The power of a power

When taking the power of a number already raised to a power, multiply the powers.

For example this is how to find the square of 23.

square of 23 = (23)2 = (2 × 2 × 2) × (2 × 2 × 2) = 26

Notice that the answer has an index of 6, which comes from multiplying the powers at the beginning (3 x 2). Here is another example.

(22)4 = (2 × 2) × (2 × 2) x (2 × 2) × (2 × 2) = 28

So you see that in both examples the powers have been multiplied (3x2 and 2x4)

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