Simplify expressions using multiple index laws with integer and variable bases and negative powers
We've already looked at a number of different index laws. Let's start by recapping these rules.
- The product rule: $a^m\times a^n=a^{m+n}$am×an=am+n
- The quotient rule: $a^m\div a^n=a^{m-n}$am÷an=am−n
- The zero index rule:$a^0=1$a0=1
- The power of a power rule: $\left(a^m\right)^n=a^{mn}$(am)n=amn
- The negative index rule: $a^{-m}=\frac{1}{a^m}$a−m=1am
We can also apply these rules to term with negative indices. The same rules apply as when we add, subtract, multiply or divide negative numbers. For example, $x^4\times x^{-9}=x^{4+\left(-9\right)}$x4×x−9=x4+(−9)$=$=$x^{-5}$x−5.
A question may have any combination of index rules. We just need to simplify it step by step, making sure we follow the order of operations.
Let's look through some examples now!
Worked Examples
Question 1
Simplify the following, giving your answer with a positive index: $2p^4q^{-2}\times5p^{-4}q^{-5}$2p4q−2×5p−4q−5
Question 2
Simplify $\left(\frac{m^7}{m^{-10}}\right)^2\times\left(\frac{m^5}{m^2}\right)^{-3}$(m7m−10)2×(m5m2)−3, giving your answer with positive indices.
Question 3
Simplify $\frac{b^3\div b^{-7}}{\left(b^{-4}\right)^{-4}}$b3÷b−7(b−4)−4, giving your answer without negative indices.