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

${(\frac{16 w^{8}}{y^{20}})}^{- \frac{3}{4}} = {(16 \times w^{8} \times \frac{1}{y^{20}})}^{- \frac{3}{4}}$

Simplify the parts where possible - in this case only $\frac{1}{y^{20}}$ can be simplified, using the law of indices " $a^{- m} = \frac{1}{a^{m}}$ ".

$\begin{array}{rcl} {(16 \times w^{8} \times \frac{1}{y^{20}})}^{- \frac{3}{4}} & = & {(16 \times w^{8} \times y^{- 20})}^{- \frac{3}{4}} \end{array}$

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

$\begin{array}{rcl} {(16 \times w^{8} \times y^{- 20})}^{- \frac{3}{4}} & = & {(16)}^{- \frac{3}{4}} \times {(w^{8})}^{- \frac{3}{4}} \times {(y^{- 20})}^{- \frac{3}{4}} \\ = & {(16)}^{- \frac{3}{4}} \times w^{- 6} \times y^{15} \end{array}$

For $w^{- 6}$ [1]
For $y^{15}$ [1]

Finally evaluate ${(16)}^{- \frac{3}{4}}$ using the law of indices " $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ " and " $a^{- m} = \frac{1}{a^{m}}$ ". Rewrite $w^{- 6}$ .

$\begin{array}{rcl} {(16)}^{- \frac{3}{4}} \times w^{- 6} \times y^{15} & = & {(\sqrt[4]{16})}^{- 3} \times \frac{1}{w^{6}} \times y^{15} \\ = & 2^{- 3} \times \frac{1}{w^{6}} \times y^{15} \\ = & \frac{1}{8} \times \frac{1}{w^{6}} \times y^{15} \\ = & \frac{y^{15}}{8 w^{6}} \end{array}$

${(\frac{16 w^{8}}{y^{20}})}^{- \frac{3}{4}} = \frac{y^{15}}{8 w^{6}}$ [1]

$2^{n + 2} + 2^{n + 1}$	$2^{2 n} + 2^{2 (n - 1)}$
$8^{n} + 8^{n - 1}$	$2^{n + 2} + 2^{n - 1}$

Algebraic Roots & Indices (AQA GCSE Maths)

Exam Questions

1. Number

Number Toolkit

Prime Factors, HCF & LCM

Powers, Roots & Standard Form

Fractions

Percentages

Simple & Compound Interest, Growth & Decay

Fractions, Decimals & Percentages

Rounding, Estimation & Bounds

Surds

Using a Calculator

2. Algebra

Algebra Toolkit

Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Algebraic Fractions

Rearranging Formulae

Algebraic Proof

Linear Equations

Solving Quadratic Equations

Simultaneous Equations

Iteration

Forming & Solving Equations

Functions

Coordinate Geometry

Linear Graphs y = mx + c

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Solving Inequalities

Graphing Inequalities

Transformations of Graphs

Sequences

3. Ratio, Proportion & Rates of Change

Ratio Toolkit

Ratio Problem Solving

Direct & Inverse Proportion

Standard & Compound Units

Exchange Rates & Best Buys

4. Geometry & Measures

Geometry Toolkit

Angles in Polygons & Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Circle Theorems

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

Congruence, Similarity & Geometrical Proof

Area & Volume of Similar Shapes

Right-Angled Triangles - Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

Vectors

Transformations

5. Probability

Probability Toolkit

Simple Probability Diagrams

Tree Diagrams

Combined & Conditional Probability

6. Statistics

Statistics Toolkit

Statistical Diagrams

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation