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KS3 Bitesize

Maths

Powers and roots

Squaring, cubing and higher powers are shown by small digits called indices, like 102 and 53.

The opposite of squaring a number is finding the square root, and the same is true for cubing and cube roots.

Introduction

This Revision Bite covers:

Powers

Nine squares

9 is a square number.
3 × 3 = 9
3 × 3 can also be written as 32. This is pronounced "3 squared".

A cube

8 is a cube number.
2 × 2 × 2 = 8
2 × 2 × 2 can also be written as 23, which is pronounced "2 cubed".

Index form

The notation 32 and 23 is known as index form. The small digit is called the index number or power.

You have already seen that 32 = 3 × 3 = 9, and that 23 = 2 × 2 × 2 = 8.

Similarly, 54 (five to the power of 4) = 5 × 5 × 5 × 5 = 625
and 35 (three to the power of 5) = 3 × 3 × 3 × 3 × 3 = 243.

The index number tells you how many times to multiply the numbers together.

  • When the index number is two, the number has been 'squared'.

  • When the index number is three, the number has been 'cubed'.

  • When the index number is greater than three you say that it is has been multiplied 'to the power of'.

For example:

72 is 'seven squared',
33 is 'three cubed',
37 is 'three to the power of seven',
45 is 'four to the power of five'.

Question

Look at the table and work out the answers. The first has been done for you.

43

4 × 4 × 4 64

27

2 × 2 × 2 × 2 × 2 × 2 × 2  

72

7 × 7  

53

   

24

   

65

   
Answer

43

4 × 4 × 4 64

27

2 × 2 × 2 × 2 × 2 × 2 × 2 128

72

7 × 7 49

53

5 × 5 × 5 125

24

2 × 2 × 2 × 2 16

65

6 × 6 × 6 × 6 × 6 7776

Powers - using a calculator

All scientific calculators have a 'power' button. This is usually labelled [xy]. This is particularly useful when the index number is large.

Example

To work out 4 10:

  • Enter 4

  • Press the xy button

  • Enter 10

  • Press =

You should get the answer 1 048 576.

Question

Use your calculator to find the values of the following:

a) 211
b) 58
c) 26 × 35

Answer

a) 2048
b) 390625
c) 26 × 35 = 64 × 243 = 15552

Square root and cube root

Square root

The opposite of squaring a number is called finding the square root.

Square root

Example

The square root of 16 is 4 (because 42 = 4 × 4 = 16)

The square root of 25 is 5 (because 52 = 5 × 5 = 25)

The square root of 100 is 10 (because 102 = 10 × 10 = 100)

Question

What is the square root of 4?

Answer

2 × 2 = 4, so 2 is the square root of 4.

The symbol '√ ' means square root, so
√ 36 means 'the square root of 36', and
√ 81 means 'the square root of 81'

You will also find a square root key on your calculator.

Cube root

The opposite of cubing a number is called finding the cube root.

Square roots

Example

The cube root of 27 is 3 (because 3 × 3 × 3 = 27)

The cube root of 1000 is 10 (because 10 × 10 × 10 = 1000)

Question

What is the cube root of 8?

Answer

2 × 2 × 2 = 8, so 2 is the cube root of 8.

Index laws

Multiplication

How can we work out 23 × 25?

23 = 2 × 2 × 2

25 = 2 × 2 × 2 × 2 × 2

so 23 × 25 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 28

There are 3 twos from 23 and 5 twos from 25, so altogether there are 8 twos.

In general, 2m × 2n =2(m + n)

Examples

25 × 24 = 2(5 + 4) = 29

27 × 23 = 2(7 + 3) = 210

The rule also works for other numbers, so

34 × 32 = 3(4 + 2) = 36

256 × 254 = 25(6 + 4) = 2510

Division

If you divide 25 by 23 you see that some of the 2's cancel:

Dividing numbers with roots or powers

shows five 2s divided by three 2s.

Five 2s are divided by three 2s

Dividing numbers with roots or powers

Shows five 2s divided by three 2s. On each side of the division, three 2s have been crossed out, leaving two 2s on the top row.

After division two 2s are left.

So 25 ÷ 23 = 22

In general, 2m ÷ 2n = 2(m - n)

Example

25 ÷ 22 = 2(5 - 2) = 23

27 ÷ 23 = 2(7 - 3) = 24

The rule also works for other numbers, so

510 ÷ 53 =5(10 - 3) = 57

459 ÷ 454 = 45(9 - 4) = 455

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