GCSE Maths

Numbers
32 Topics | 1 Quiz
Integers
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
29 Topics | 1 Quiz
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
7 Topics
Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
15 Topics
2D Shapes
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
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
6 Topics
Introduction to probability
Finding the sum of probabilities
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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The Laws of Indices

GCSE Maths Numbers The Laws of Indices

Introduction to Indices

An index or power is a small number placed above and to the right of a base number. The index indicates how many times the base number should be multiplied by itself.

For example, 2\times 2\times 2\times 2\times 2 can be written as { 2 }^{ 5 }. The base is 2, and the exponent is 5. It is read as “2 to the power of 5.”

Let’s look at some more examples:

  • 3\times 3={ 3 }^{ 2 }
  • 5\times 5\times 5\times 5={ 5 }^{ 4 }
  • 7\times 7\times 7={ 7 }^{ 3 }

Laws of Indices

Multiplying indices

When multiplying indices with the same base, we add the powers:

a^{b}\times a^{c}=a^{b+c}

Examples:

  • 3^{ 7 }\times { 3 }^{ 4 }=3^{ 7+4 }={ 3 }^{ 11 }
  • { 5 }^{ 8 }\times { 5 }^{ 2 }={ 5 }^{ 10 }
  • { 3 }^{ 3 }\times { 3 }^{ 6 }={ 3 }^{ 9 }

Dividing indices

When dividing indices with the same base, we subtract the powers:

\dfrac{a^{b}}{a^{c}}=a^{b-c}

Examples:

  • { 8 }^{ 14 }\div{ 8 }^{ 7 }={ 8 }^{ 14-7 }={ 8 }^{ 7 }
  • { 2 }^{ 10 }\div{ 2 }^{ 4 }={ 2 }^{ 10-4 }={ 2 }^{ 6 }
  • { 3 }^{ 7 }\div { 3 }^{ 5 }={ 3 }^{ 7-5 }={ 3 }^{ 2 }
  • { 5 }^{ 20 }\div { 5 }^{ 12 }={ 5 }^{ 20-12 }={ 5 }^{ 8 }

Power of a power

When raising an index to another power, we multiply the powers:

\left( a^{b}\right) ^{c}=a^{b\times c}

Examples:

  • ({ { 3 }^{ 2 }) }^{ 4 }={ 3 }^{ 2 }\times { 3 }^{ 2 }\times { 3 }^{ 2 }\times { 3 }^{ 2 }={ 3 }^{ 2+2+2+2 }=3^{8}
  • ({ { 2 }^{ 5 }) }^{ 2 }={ 2 }^{ 5\times 2 }={ 2 }^{ 10 }
  • ({ { 3 }^{ 7 }) }^{ 2 }={ 3 }^{ 7\times 2 }={ 3 }^{ 14 }
  • ({ { 5 }^{ 10 }) }^{ 10 }={ 5 }^{ 10\times 10 }={ 5 }^{ 100 }
  • ({ { 4 }^{ 3 }) }^{ 5 }={ 4 }^{ 3\times 5 }={ 4 }^{ 15 }

Power of a product

When raising a product to a power, we distribute the power to each factor:

\left( ab\right) ^{n}=a^{n}\times b^{n}

Examples:

\left( 4\times 3\right) ^{2}=4^{2}\times 3^{2}

\left( 5\times 7\right) ^{3}=5^{3}\times 7^{3}

Power of zero

Any non-zero number raised to the power of 0 is equal to 1:

a^{0}=1 (where a\neq 0)

Examples:

  • { 5 }^{ 0 }=1
  • { 8 }^{ 0 }=1
  • { 1000 }^{ 0 }=1,\quad ({ -3) }^{ 0 }=1

Negative indices

A negative index represents the reciprocal of the positive index:

a^{-n}=\dfrac{1}{a^{n}}

Examples:

  • 2^{-3}=\dfrac{1}{2^{3}}=\dfrac{1}{8}
  • 4^{-4}=\dfrac{1}{4^{4}}=\dfrac{1}{256}

Using the Laws of Indices to Simplify Numbers and Equations

We can use the laws of indices to simplify numbers and calculations. For example:

If A=2.3\times 10^{5}, B=4.5\times 10^{-2}, then AB=\left( 2.3\times 10^{5}\right) \times \left( 4.5\times 10^{-2}\right)

=10.35\times 10^{5-2}

=10.35\times 10^{3}

=1.035\times 10\times 10^{3}=1.035\times 10^{4}

Let’s look at two more examples:

Example

Simplify:

1. 3^{5}\times 3^{2}

2. 3^{5}\div 3^{3}

3. \left( 3^{2}\right) ^{5}

4. 3^{-4}

5. 3^{0}

6. \left( 3\times 7\right) ^{2}

1. 3^{5}\times 3^{2}=3^{5+2}=3^{7}

2. 3^{5}\div 3^{3}=3^{5-3}=3^{2}=9

3. \left( 3^{2}\right) ^{5}=3^{2\times 5}=3^{10}

4. 3^{-4}=\dfrac{1}{3^4}=\dfrac{1}{81}

5. 3^{0}=1

6. \left( 3\times 7\right) ^{2}=3^{2}\times 7^{2}=9\times 49=441

Example

Simplify \left( 5^{3}\right) ^{2}\times \left( 5^{-1}\right) ^{4}

\left( 5^{3}\right) ^{2}\times \left( 5^{-1}\right) ^{4}=5^{6}\times 5^{-4}

=5^{6-4}

=5^{2}

=25

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