Division law with integer or variable bases and integer negative powers
We've already learnt about the division rule which states:
$a^x\div a^y=a^{x-y}$ax÷ay=ax−y
What happens when we encounter questions that include negative indices? The same rule applies. We just need to remember the rule for working with negative numbers.
-
Subtracting a negative term is the same as adding a positive term, e.g. $2-\left(-5\right)=2+5$2−(−5)=2+5$=$=$7$7
-
Adding a negative term is the same as subtracting the term, e.g. $6+\left(-10\right)=6-10$6+(−10)=6−10$=$=$-4$−4
Since we are working with directed numbers, it is important the we're also familiar with the negative index law, which states $a^{-x}=\frac{1}{a^x}$a−x=1ax, just in case we're asked to express a negative index and a positive index or vice versa.
Examples
Question 1
Rewrite $10^{-10}\div10^4$10−10÷104 in the form $a^n$an.
Question 2
Simplify the following, giving your answer with positive indices: $\frac{5p^5q^{-4}}{40p^5q^6}$5p5q−440p5q6
Question 3
Rewrite the following expression using positive index notation with a single base: $\frac{1}{9\times9\times9\times9}\times\frac{1}{9\times9}$19×9×9×9×19×9.