We've already looked at a number of different exponent 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 exponent rule:$a^0=1$a0=1
- The power of a power rule: $\left(a^m\right)^n=a^{mn}$(am)n=amn
- The negative exponent rule: $a^{-m}=\frac{1}{a^m}$a−m=1am
We can also apply these rules to term with negative exponents. 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 exponent 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 exponent: $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 exponents.
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 exponents.