5.03 Negative indices
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 $a^{-2}$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.
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
Express $6^{-10}$6−10 with a positive index.
Question 2 (Extended)
Simplify $\frac{\left(5^2\right)^9\times5^6}{5^{40}}$(52)9×56540, giving your answer in the form $a^n$an.