5.03 Negative indices
Negative indices
So far we have looked at expressions of the form $$aman where $$m>n and where $$m=n, and how to simplify them using the division rule and also the zero power rule.
But what happens when $$m is smaller than $$n? For example, if we simplified $$a3÷a5 using the division law, we would get $$a−2. So what does a negative index mean? Let's expand the example to find out:
Remember that when we are simplifying fractions, we are looking to cancel out common factors in the numerator and denominator. Remember that any number divided by itself is $$1.
So using the second approach, we can also express $$a3÷a5 with a positive index as $$1a2. The result is summarised by the negative index law.
For any base $$a,
$$a−x=1ax, $$a≠0.
That is, when raising a base to a negative power:
- Take the reciprocal of the expression
- Turn the power into a negative
Practice questions
Question 1
Express $$6−10 with a positive index.
Question 2 (Extended)
Simplify $$(52)9×56540, giving your answer in the form $$an.