First separate numbers and letters in the fraction inside the brackets.

${(\frac{9 t^{4} w^{9}}{18 t^{6} w^{10}})}^{- 2} = {(\frac{9}{18} \times \frac{t^{4}}{t^{6}} \times \frac{w^{9}}{w^{10}})}^{- 2}$

Simplify the numerical fraction and use the law of indices " $a^{m} \div a^{n} = a^{m - n}$ " to simplify the algebraic parts.

$\begin{array}{rcl} {(\frac{9}{18} \times \frac{t^{4}}{t^{6}} \times \frac{w^{9}}{w^{10}})}^{- 2} & = & {(\frac{1}{2} \times t^{4 - 6} \times w^{9 - 10})}^{- 2} \\ = & {(\frac{1}{2} \times t^{- 2} \times w^{- 1})}^{- 2} \end{array}$

Use the laws of indices " ${(a b)}^{n}$ " and " ${(a^{m})}^{n} = a^{m n}$ " to deal with the power of -2 at the end.

$\begin{array}{rcl} {(\frac{1}{2} \times t^{- 2} \times w^{- 1})}^{- 2} & = & {(\frac{1}{2})}^{- 2} \times {(t^{- 2})}^{- 2} \times {(w^{- 1})}^{- 2} \\ = & {(\frac{1}{2})}^{- 2} \times t^{4} \times w^{2} \end{array}$

[1]

Finally evaluate ${(\frac{1}{2})}^{- 2}$ by writing $\frac{1}{2}$ as $2^{- 1}$ and write your final answer as simply as possible.
Use your calculator if unsure for this part.

$\begin{array}{rcl} {(\frac{1}{2})}^{- 2} \times t^{4} \times w^{2} & = & {(2^{- 1})}^{- 2} \times t^{4} \times w^{2} \\ = & 2^{2} \times t^{4} \times w^{2} \\ = & 4 t^{4} w^{2} \end{array}$

Correctly applying the "-2" outside power [1]
${(\frac{9 t^{4} w^{9}}{18 t^{6} w^{10}})}^{- 2} = 4 t^{4} w^{2}$ [1]

Algebraic Roots & Indices (CIE IGCSE Maths: Extended)

Exam Questions