 iGCSE (2021 Edition)

1.03 Negative indices

Lesson

Negative indices

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 $a^{-2}$a2. 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,

$a^{-x}=\frac{1}{a^x}$ax=1ax$a\ne0$a0.

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

Express $6^{-10}$610 with a positive index.

Question 2

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

Outcomes

0606C4.1A

Perform simple operations with indices.