GCSE Maths

Numbers
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Integers
Integers Quiz
Prime Numbers and Composite Numbers
Positive and Negative Numbers
Order of Operations
Square Numbers and Cube Numbers
The Laws of Indices
Square Roots and Cube Roots
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
Expressing Fractions as Decimal Numbers
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Algebra
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Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Substituting Values into Formulae
Rearranging Equations
Algebraic Fractions
Expanding Brackets and Simplifying
Expanding Double Brackets
Factorisation
Factorising Quadratic Expressions
Factorising Quadratics with a Coefficient of x Squared Greater than 1
The Quadratic Formula
The Difference of Two Squares
Subject of Formula
Solving Linear Equations
Quadratic Equations
Completing the Square
Sequences and Nth Term
Arithmetic and Geometric Sequences
Quadratic Sequences
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Ratio, Proportion and Rates of Change
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Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
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2D Shapes
2D Shapes Quiz
3D Shapes
3D Shapes Quiz
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
Triangles Quiz
Angles in Triangles
Polygons
Lines of Symmetry
Bearings
Loci and constructions
Constructing Triangles
Transformations
Enlargement
Plans and elevations
Areas and Volumes in Map Scale Problems
Probability
10 Topics
Introduction to Probability
Relative Frequency
Sample Space
Probability of Single Events
Probability of Combined Events
Probability Trees
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Vectors
Mensuration and Calculation
7 Topics
Area
Volume
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Fractional Indices

GCSE Maths Numbers Fractional Indices

Fractional indices can be expressed as roots. Here are some basic rules:

  • a^{\frac{1}{2}}=\sqrt{a} (square root)
  • a^{\frac{1}{3}}=\sqrt[3]{a} (cube root)
  • a^{\frac{1}{n}}=\sqrt[n]{a} (nth root)
  • a^{\frac{m}{n}}=\left(a^{m}\right)^{\frac{1}{n}}=\sqrt[n]{a^{m}}

Examples

Simplify the following expressions:

1. 8^{\frac{2}{3}}

2. 16^{\frac{1}{4}}

3. 125^{-\frac{2}{3}}

4. 64^{-\frac{4}{3}}

5. 36^{-\frac{1}{2}}

6. 81^{\frac{5}{4}}

7. \left(\frac{2}{3}\right)^{-2}

8. -8^{\frac{2}{3}}

9. -27^{-\frac{4}{3}}

1. 8^{\frac{2}{3}}=\left(2^{3}\right)^{\frac{2}{3}}=2^{2}=4

2. 16^{\frac{1}{4}}=\left(2^{4}\right)^{\frac{1}{4}}=2^{1}=2

3. 125^{-\frac{2}{3}}=\left(5^{3}\right)^{-\frac{2}{3}}=5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}

4. 64^{-\frac{4}{3}}=\left(2^{6}\right)^{-\frac{4}{3}}=2^{-4}=\frac{1}{2^{4}}=\frac{1}{16}

5. 36^{-\frac{1}{2}}=\left(6^{2}\right)^{-\frac{1}{2}}=6^{-1}=\frac{1}{6}

6. 81^{\frac{5}{4}}=\left(3^{4}\right)^{\frac{5}{4}}=3^{5}=243

7. \left(\frac{2}{3}\right)^{-2}=\frac{1}{\left(\frac{2}{3}\right)^{2}}=\frac{1}{\frac{4}{9}}=1\div \frac{4}{9}=1\times \frac{9}{4}=\frac{9}{4}

Note that, \left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^{m}

8. -8^{\frac{2}{3}}=\left[\left(-2\right)^{3}\right]^{\frac{2}{3}}=\left(-2\right)^{2}=4

9. -27^{-\frac{4}{3}}=\left[\left(-3\right)^{3}\right]^{-\frac{4}{3}}=\left(-3\right)^{-4}=\frac{1}{\left(-3\right)^{4}}=\frac{1}{81}

Example

Evaluate the following:

1. 16^{-\frac{3}{4}}

2. 125^{-\frac{1}{3}}

3. \left(\frac{9}{16}\right)^{-\frac{3}{2}}

4. \left(2\frac{1}{4}\right)^3

5. \left(2\frac{1}{4}\right)^{-3}

6. \left(2\frac{1}{4}\right)^{\frac{3}{2}}

7. \left(2\frac{1}{4}\right)^{-\frac{3}{2}}

1. 16^{-\frac{3}{4}}=\left(2^{4}\right)^{-\frac{3}{4}}=2^{-3}=\frac{1}{2^3}=\frac{1}{8}

2. 125^{-\frac{1}{3}}=\left(5^{3}\right)^{-\frac{1}{3}}=5^{-1}=\frac{1}{5}

3. \left(\frac{9}{16}\right)^{-\frac{3}{2}}=\left(\left(\frac{3}{4}\right)^2\right)^{-\frac{3}{2}}=\left(\frac{3}{4}\right)^{-3}=\frac{1}{\left(\frac{3}{4}\right)^3}=\frac{1}{\frac{27}{64}}=\frac{64}{27}

4. \left(2\frac{1}{4}\right)^3=\left(\frac{9}{4}\right)^3=\frac{729}{64}

5. \left(2\frac{1}{4}\right)^{-3}=\left(\frac{9}{4}\right)^{-3}=\left(\frac{4}{9}\right)^3=\frac{64}{729}

6. \left(2\frac{1}{4}\right)^{\frac{3}{2}}=\left(\frac{9}{4}\right)^{\frac{3}{2}}=\left(\left(\frac{3}{2}\right)^2\right)^{\frac{3}{2}}=\left(\frac{3}{2}\right)^3=\frac{27}{8}

7. \left(2\frac{1}{4}\right)^{-\frac{3}{2}}=\left(\frac{9}{4}\right)^{-\frac{3}{2}}=\left(\left(\frac{3}{2}\right)^2\right)^{-\frac{3}{2}}=\left(\frac{3}{2}\right)^{-3}=\left(\frac{2}{3}\right)^3=\frac{8}{27}

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