Multiplication law with integer or variable bases and negative powers
In Multiplying Multiple Powers, we looked at the multiplication law, which states:
$a^x\times a^y=a^{x+y}$ax×ay=ax+y
The base terms needs to be the same to apply this rule.
We can also apply the multiplication rule to terms with negative indices. We just need to remember how to add with negative numbers.
Also, we may be asked to express terms with negative indices with positive indices instead. To do this, we need to apply the negative index rule, which states:
$a^{-x}=\frac{1}{a^x}$a−x=1ax
Let's look through some examples now to see these processes in action.
Examples
Question 1
Rewrite $10^{-10}\times10^4$10−10×104 in the form $a^n$an.
Question 2
Simplify the following, giving your answer with a positive index: $p^2\times p^{-7}$p2×p−7
Question 3
Simplify the following, writing without negative indices.
$2p^5q^{-7}\times6p^{-9}q^9$2p5q−7×6p−9q9