Laws of indices

In a nutshell

Indices, or an index, refers to the power on a number or variable. There are different rules which show how to manipulate expressions with indices, including how to multiply or divide indices or how to calculate bracketed indices. This set of rules is known as the index laws.

Multiplying with indices

If two numbers with powers are multiplied together, it is possible to simplify the power, as long as the bases of the two numbers are the same. When two numbers with the same base, but different powers are multiplied, add the powers. Use the rule:

$\boxed{a^m \times a^n = a^{m+n}}$

Example 1

Simplify $2x^5 \times 3x^3$ .

The terms are multiplied, so add the powers.

$\begin {aligned}2x^5 \times 3x^3 &= 6x ^ {5+3} \\&= \underline{6x^8}\end {aligned}$

Dividing with indices

If two numbers with powers are divided, it is possible to simplify the power, as long as the bases of the two numbers are the same. When two numbers with the same base, but different powers are divided, subtract the powers. Use the rule:

$\boxed{a^m \div a^n = a^{m-n}}$

Example 2

Simplify $15x^5 \div 3x^2$ .

The terms are divided, so subtract the powers.

$\begin {aligned}15x^5 \div 3x^2 &= 5x ^ {5-2} \\&= \underline{5x^3}\end {aligned}$ 

Bracketed indices

Bracketed indices are expressions where a term with a power is raised to another power, for example $(x^3)^4$ . Here, $x^3$ is multiplied by itself four times. Bracketed indices can be simplified by multiplying the powers. Use the rule:

$\boxed{(a^m)^n = a^{mn}}$

Example 3

Simplify $(2x^3)^2$ .

Multiply the indices.

$\begin{aligned}(2x^3)^2 &= 2^2\times x^{3 \times 2} \\&= \underline{4x^6}\end {aligned}$

Tip: Remember that the coefficient is also raised to the power!