1. Indices and Surds

iGCSE (2021 Edition)

Lesson

So far we have looked at expressions of the form $\frac{a^m}{a^n}$`a``m``a``n` 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 $a^3\div a^5$`a`3÷`a`5 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 $a^3\div a^5$`a`3÷`a`5 with a positive index as $\frac{1}{a^2}$1`a`2. 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`=1`a``x`, $a\ne0$`a`≠0.

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

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

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

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

Perform simple operations with indices.