So far we have looked at expressions of the form $\frac{a^m}{a^n}$aman where $m>n$m>n and where $m=n$m=n, and how to simplify them using the division rule and also the zero power rule.

But what happens when $m$m is smaller than $n$n? For example, if we simplified $a^3\div a^5$a3÷a5 using the division law, we would get $a^{-2}$a−2. So what does a negative index mean? Let's expand the example to find out:

Remember that when we are simplifying fractions, we are looking to cancel out common factors in the numerator and denominator. Remember that any number divided by itself is $1$1.

So using the second approach, we can also express $a^3\div a^5$a3÷a5 with a positive index as $\frac{1}{a^2}$1a2. The result is summarised by the negative index law.

Negative index law

For any base $a$a,

$a^{-x}=\frac{1}{a^x}$a−x=1ax, $a\ne0$a≠0.

That is, when raising a base to a negative power:

Take the reciprocal of the expression
Turn the power into a negative

Practice questions

Question 1

Evaluate $\left(6\times19\right)^0$(6×19)0.

Question 2

Express $6^{-10}$6−10 with a positive index.

Question 3

Simplify $\frac{\left(5^2\right)^9\times5^6}{5^{40}}$(52)9×56540, giving your answer in the form $a^n$an.

Outcomes

0580C1.7A

Understand the meaning of indices (fractional, negative and zero) and use the rules of indices.

0580E1.7A

Understand the meaning of indices (fractional, negative and zero) and use the rules of indices.

0580C2.4

Use and interpret positive, negative and zero indices. Use the rules of indices.

0580E2.4A