Indices

You can find questions, worked solutions and a crib sheet at the bottom of the page.

Indices tell us how many times to multiply a number. It is also known as a power. For example,

23 = 2 × 2 × 2

For your exam there are several “Rules of Indices” you need to remember.

The Rules of Indices

First rule:

am × an = am+n

If we have the product (multiplication) of the same number with different powers then we add the powers.

For example:

x2 × x3 = x2+3 = x5

22 × 23 = 22+3 = 25 = 32

But the subject number (a in the rule) must be the same in both cases. We cannot combine the powers of two different numbers.

32 × 23 ≠ 22+3 ≠ 32+3 ≠ 62+3

Second rule:

am ÷ an = am-n

This is similar to the first rule but if we divide, we subtract the powers.

For example:

x5 ÷ x3 = x5-3 = x2

45 ÷ 43 = 45-3 = 42 = 16

But as before the subject number (a in the rule) must be the same in both cases. We cannot combine the powers of two different numbers.

Third rule:

(am)n = am×n = amn

If we have a power of a power (like the cube of a number squared) we multiply the powers together.

For example:

(x2)3 = x2×3 = x6

(22)3 = 22×3 = 26 = 64

Fourth rule:

a-1 = 1a            or more generally            a-m = 1 ⁄ am

A number to the power of -1 becomes one over that number (a fraction). Using the third rule, it makes sense that a number to the power of -2 becomes one over that number squared and a number to the power of -3 is one over that number cubed and so on.

For example:


x-1 = 1⁄ x

3-1 = 13

x-2 = 1 ⁄ x2

3-2 = 1 ⁄ 32 = 19

Fifth rule:

a1/2 = √a          or more generally          a1/n = n√a

A number to the power of 1 over n is equal to the nth root of that number.

For example:

x1/2 = √x

91/2 = √9 = 3

x1/3 = 3√x

81/3 = 3√8 = 2

82/3 = (3√8) 2 = ( 2 )2 = 4

Those are the main 5 rules but there are two more things to be aware of:

x0 = 1

1a = 1

234320 = 1

1237 = 1

Anything to the power of zero is equal to 1.

1 to the power of anything is equal to 1.


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