# Laws of indices

## Introduction

You need to be familiar with these laws of indices.

$$x^a × x^b = x^{a + b}$$.

$$x^a ÷ x^b = x^{a - b}$$.

$$(x^a)^b = x^{a × b}$$.

Learn these special cases off by heart.

Anything to the power of 1 is itself, i.e. $$x^{1} = x$$.

Anything to the power of 0 is 1, i.e. $$x^{0} = 1$$.

$$x^{-1} = \frac{1}{x}$$.

## How to apply the laws of indices

### Example 1 - Multiplying powers

a) Simplify $$5^6 × 5^7$$.

When multiplying, add together the powers:

\begin{aligned} 5^6 × 5^7 & = 5^{6 + 7}\\ & = 5^{13}. \end{aligned}

b) Simplify $$3a^2 × 4a^7$$.

Multiply the numbers as normal, then add together the powers:

\begin{aligned} 3x^2 × 4x^7 & = 12x^{2 + 7}\\ & = 12x^9. \end{aligned}

### Example 2 - Dividing powers

Simplify $$x^8 ÷ x^2$$.

When dividing, subtract the powers:

\begin{aligned} x^8 ÷ x^2 & = x^{8 - 2}\\ & = x^6. \end{aligned}

### Example 3 - Raising a power to another power

Simplify $$(x^2)^6$$.

When raising a power to another power, multiply the powers:

\begin{aligned} (x^2)^6 & = x^{2 × 6}\\ & = x^{12}. \end{aligned}

## Worksheets to practise applying the basic laws of indices

Try this worksheet to practise your skills.