GCSE Maths

Numbers
38 Topics | 1 Quiz
Integers
Prime Numbers
Composite Numbers
Negative Numbers
Comparing Numbers Using Symbols
The Square Numbers
The Laws of Indices
Negative Indices
The Cube Numbers
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
1 of 2
Vectors
1 Topic
Vectors
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
Algebra
16 Topics | 1 Quiz
Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Expanding Brackets and Simplifying
Expanding Double Brackets
The Difference of Two Squares
Substituting into Formulae
Subject of Formula
Factorisation
Factorising Quadratic Expressions
More on Factorisation
Equations
Quadratic Equations
Mathematical Formulae
Algebraic Equivalence and Proof
Functions
Algebra
Sequences
2 Topics | 1 Quiz
Generating sequences
Nth Term Sequences
Sequences
Ratio, Proportion and Rates of Change
8 Topics | 1 Quiz
Inverse Proportion
Conversion
Scale Factors
Percentages
Direct Proportion
Rate, Ratio and Proportion Examples
Ratios
Ratios
Ratio, Proportion and Rates of Change
Geometry and Measures
9 Topics
Loci and constructions
Properties of angles
Properties of polygons
Transformations
Enlargement
Circle theorems
Plane figure properties
Plans and elevations
Areas and Volumes in Map Scale Problems
Mensuration and Calculation
10 Topics
Map scales
Bearings
Area
Volume
Circles: arcs and sectors
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

We write 2\times 2\times 2\times 2\times 2 as { 2 }^{ 5 }. The small 5 on the top is the number of times 2 appears. We read it as 2 to the power of 5. The power is 5 and 2 is the base.

Likewise, 3\times 3={ 3 }^{ 2 }

5\times 5\times 5\times 5={ 5 }^{ 4 }

7\times 7\times 7={ 7 }^{ 3 }

Laws of indices

Suppose we wish to simplify { 2 }^{ 5 }\times { 2 }^{ 3 }

We can write this as ({ 2 }\times 2\times 2\times 2\times 2)\times (2\times 2\times 2).

But this is the same as { 2 }\times 2\times 2\times 2\times 2\times 2\times 2\times 2

The “2” appears 8 times. Therefore { 2 }^{ 5 }\times { 2 }^{ 3 }={ 2 }^{ 8 }

Multiplying indices

When we are multiplying indices, we add the powers, as long as the base is the same.

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

For example:

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 we are dividing indices, we subtract powers, as long as the base is the same.

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

For example:

{ 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 }

The rules for brackets

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

For example:

({ { 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 3 }={ 2 }^{ 15 }

({ { 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 }

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

For example:

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

When the power is zero

Except from 0, any number raised to the power of 0 is 1.

For example:

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

Negative indices

We can also call a negative power a reciprocal

a^{-n}=\dfrac{1}{a^{n}} is the reciprocal of a^{n}

For example:

2^{-3}=\dfrac{1}{2^{3}}=\dfrac{1}{8}

4^{-4}=\dfrac{1}{4^{4}}=\dfrac{1}{256}

Using the Law 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}

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}

Solutions:

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}=3\dfrac{1}{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}

Solution:

\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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