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:

a^m × aⁿ = a^m+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:

a^m ÷ aⁿ = a^m-n

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

For example:

x⁵ ÷ x³ = x^5-3 = x²

4⁵ ÷ 4³ = 4^5-3 = 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:

(a^m)ⁿ = a^m×n = a^mn

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

For example:

(x²)³ = x^2×3 = x⁶

(2²)³ = 2^2×3 = 2⁶ = 64

Fourth rule:

a^-1 = ¹⁄_a or more generally a^-m = ¹ ⁄ a^m

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 = ¹⁄₃

x^-2 = ¹ ⁄ x²

3^-2 = ¹ ⁄ 3² = ¹⁄₉