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Mixed expressions using index laws


We've learnt a number of indice rules. Now we are going to look at questions that involve a combination of these rules. It's important to remember the order of operations when we're solving these questions.

Rule Recap
  • 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=amn
  • 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: $a^{-m}=\frac{1}{a^m}$am=1am

A question may have any combination of indice 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!


Question 1

Simplify: $p^7\div p^3\times p^5$p7÷​p3×p5

Think: We need to apply the index division and index multiplication laws.


$p^7\div p^3\times p^5$p7÷​p3×p5 $=$= $p^{7-3+5}$p73+5
  $=$= $p^9$p9


Question 2

Simplify: $\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(ux+3)3ux+1

Think: Just like in Question 2, we need to simplify the numerator using the power of a power rule, then apply the quotient rule.


$\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(ux+3)3ux+1 $=$= $\frac{u^{3x+9}}{u^{x+1}}$u3x+9ux+1 Firstly, we'll simplify the numerator using the "power of a power" rule
  $=$= $u^{3x+9-\left(x+1\right)}$u3x+9(x+1) Then, using the quotient rule, we can subtract the power
  $=$= $u^{3x+9-x-1}$u3x+9x1 Expand the brackets, then simplify by collecting the like terms
  $=$= $u^{2x+8}$u2x+8  


Question 3

Express $\left(4^p\right)^4$(4p)4 with a prime number base in index form.

Think: We could express $4$4 as $2^2$22 which has a prime number base.


$\left(4^p\right)^4$(4p)4 $=$= $4^{4p}$44p
  $=$= $\left(2^2\right)^{4p}$(22)4p
  $=$= $2^{8p}$28p


Question 4

Simplify $20m^6\div5m^{13}\times9m^2$20m6÷​5m13×9m2, expressing your answer in positive index form.

Think: Let's express this as a fraction so the powers are on the numerator and the denominator for easy comparison.


$\frac{20m^6}{5m^{13}}\times9m^2$20m65m13×9m2 $=$= $\frac{4}{m^7}\times9m^2$4m7×9m2
  $=$= $\frac{36m^2}{m^7}$36m2m7
  $=$= $\frac{36}{m^5}$36m5
Question 5

Simplify $\frac{\left(x^2\right)^6}{\left(x^2\right)^2}$(x2)6(x2)2

Question 6

Simplify $\left(u^9\times u^5\div u^{19}\right)^2$(u9×u5÷​u19)2, expressing your answer in positive index form.


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