Rationalise Denominator

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Rationalising the Denominator

It is often easier to work with surds when there are no square roots on the bottom of a fraction. Removing surds from the bottom of a fraction is called ‘rationalising the denominator’.

Simple example

Advanced example

To rationalise a denominator of the form a ± √b multiply by the denominator but with the sign in front of the root changed.

Exact form

Leaving an answer in exact form means leaving any fractions, surds and constants like in the expression rather than giving the answer as a decimal.

Number

1.1

Place Value & Ordering

1.1.1Place Value

1.2

Numerical Operations

1.2.1Adding Integers 1.2.2Subtracting Integers 1.2.3Multiplying Integers 1.2.4Dividing Integers 1.2.5Adding & Subtracting Decimals 1.2.6Multiplying Decimals 1.2.7Dividing Decimals 1.2.8Adding & Subtracting Negative Numbers 1.2.9Multiplying & Dividing Negative Numbers 1.2.10Multiplying by Powers of Ten 1.2.11Dividing by Powers of Ten

1.3

Structure of Calculations

1.3.1Order of Operations (BIDMAS)

1.4

Fractions

1.4.1Writing Fractions 1.4.2Simplifying Fractions 1.4.3Fractions & Recurring Decimals

1.5

Numerical Operations on Fractions

1.5.1Adding & Subtracting Fractions 1.5.2Adding & Subtracting Mixed Numbers 1.5.3Multiplying Fractions 1.5.4Multiplying Mixed Numbers 1.5.5Dividing Fractions 1.5.6Dividing Mixed Numbers (sample)1.5.7Dividing Mixed Numbers

1.6

Roots & Powers

1.6.1Roots 1.6.2Fractional Indices (1/n)1.6.3Fractional Indices (m/n)1.6.4Laws of Indices (Multiplication)1.6.5Laws of Indices (Division)

1.7

Numerical Skills

1.7.1Using a Calculator - Powers, Roots & Fractions 1.7.2Using a Calculator - Trigonometric Functions

1.8

Factors & Multiples

1.8.1Finding Factors 1.8.2Finding Multiples

1.9

Surds

1.9.1Multiplying & Dividing Surds 1.9.2Simplifying Surds 1.9.3Expanding Expressions With Surds 1.9.4Rationalising the Denominator

1.10

Standard Form

1.10.1Converting to Standard Form (Positive Powers)1.10.2Converting to Standard Form (Negative Powers)1.10.3Converting From Standard Form (Positive Powers)1.10.4Converting From Standard Form (Negative Powers)1.10.5Adding & Subtracting Standard Form 1.10.6Multiplying & Dividing Standard Form

1.11

Fractions, Decimals & Percentages

1.11.1Converting Fractions, Decimals & Percentages 1.11.2Converting Recurring Decimals 1.11.3Numbers as Percentages 1.11.4Fractions of Amounts

1.12

Percentage Change

1.12.1Percentages of Amounts (No Calculator)1.12.2Percentage Change 1.12.3Original Values After Percentage Change 1.12.4Growth & Decay Using Multipliers

1.13

Rounding & Estimation

1.13.1Rounding Integers 1.13.2Estimating Calculations 1.13.3Rounding Decimals 1.13.4Rounding With Significant Figures

1.14

Bounds & Accuracy

1.14.1Finding Error Intervals 1.14.2Upper & Lower Bounds 1.14.3Error Intervals From Truncation

1.15

Using Numbers

1.15.1Equal Symbols 1.15.2Negative Numbers 1.15.3BIDMAS 1.15.4Inverse Operations 1.15.5End of Topic Test- Ordering 1.15.6Factors 1.15.7Multiples 1.15.8Highest Common Factor 1.15.9Lowest Common Multiple 1.15.10Number of Ways to Do a Task 1.15.11Number of Combinations 1.15.12Number of Ways To Arrange Objects 1.15.13End of Topic Test - Factors, Multiples & Combos 1.15.14Standard Form 1.15.15Ordinary Numbers to Standard Form 1.15.16Addition in Standard Form 1.15.17Subtraction in Standard Form 1.15.18Multiplication in Standard Form 1.15.19Division in Standard Form 1.15.20Exam-Style Questions - Standard Form & Primes 1.15.21Long Division 1.15.22Prime Numbers 1.15.23End of Topic Test - Standard Form & Long Division 1.15.24Diagnostic Misconceptions - 1 NOT a Prime Number 1.15.25Diagnostic Misconceptions - BIDMAS 1.15.26Diagnostic Misconceptions - Adding Negatives

1.16

Fractions, Decimals & Percentages

1.16.1Adding Fractions 1.16.2Subtracting Fractions 1.16.3Multiplying Fractions 1.16.4Dividing Fractions 1.16.5Mixed Fractions 1.16.6Calculating Percentages of Numbers 1.16.7Calculating Fractions of Numbers 1.16.8Multiply Decimals 1.16.9Place Value 1.16.10Recurring Decimals 1.16.11Exam-Style Questions - Recurring Decimals 1.16.12Convert Fractions & Terminating Decimals 1.16.13Convert Recurring Decimals 1.16.14Convert Decimals & Percentages 1.16.15Compare Decimals 1.16.16Compare Fractions 1.16.17Exam-Style Questions - Percentages 1.16.18End of Topic Test - Fractions & Decimals 1.16.19Diagnostic Misconceptions - Adding Fractions

1.17

Powers & Roots

1.17.1Powers 1.17.2Roots 1.17.3Indices Advanced 1.17.4Estimating Powers 1.17.5Estimating Roots 1.17.6Multiplying Powers 1.17.7Dividing Powers 1.17.8Power Raised to Power 1.17.9Fractions Raised to a Power 1.17.10Numbers Raised to Negative Powers 1.17.11Roots of Numbers 1.17.12Numbers to the Power of 1 & 0 1.17.13Surds 1.17.14Simplify Surds 1.17.15Rationalise Denominator 1.17.16Exam-Style Questions - Indices 1.17.17End of Topic Test - Powers & Roots 1.17.18Diagnostic Misconceptions - Squaring 1.17.19Diagnostic Misconceptions - Simplifying Surds

1.18

Accuracy

1.18.1Decimal Rounding 1.18.2Significant Figure Rounding 1.18.3Estimation 1.18.4Truncation 1.18.5Limits of Accuracy 1.18.6Bounds Addition 1.18.7Bounds Subtraction 1.18.8Bounds Multiplication 1.18.9Bounds Division 1.18.10End of Topic Test - Rounding & Estimation 1.18.11Diagnostic Misconceptions - Upper Bounds

Algebra

2.1

Algebraic Notation

2.1.1Using Algebraic Notation

2.2

Substitution & Simplification

2.2.1Substituting into Expressions 2.2.2Substituting into Real-Life Formulae 2.2.3Simplifying Expressions by Collecting Like Terms

2.3

Brackets

2.3.1Expanding Single Brackets 2.3.2Expanding Double Brackets 2.3.3Factorising into One Bracket 2.3.4Factorising Quadratics of the Form x^2 + bx + c 2.3.5Factorising Quadratics - AC Method 2.3.6Factorising Quadratics - Middle Term Method 2.3.7Factorising the Difference of Two Squares 2.3.8Completing the Square

2.4

Algebraic Fractions

2.4.1Simplifying Algebraic Fractions by Cancelling 2.4.2Simplifying Algebraic Fractions by Factorising 2.4.3Operations with Algebraic Fractions

2.5

Rearranging

2.5.1Changing the Subject 2.5.2Changing the Subject (Repeated Subject)

2.6

Functions

2.6.1Finding the Input of a Function 2.6.2Inverse Functions

2.7

Solving Linear Equations

2.7.1Solving Linear Equations (One Step)2.7.2Solving Linear Equations (Multi-Step)2.7.3Linear Equations (Unknown Both Sides)2.7.4Solving Equations: (Unknown Denominator)2.7.5Forming & Solving Linear Equations

2.8

Inequalities

2.8.1Solving Linear Inequalities 2.8.2Linear Inequalities with Unknowns on Both Sides

2.9

Solving Quadratic Equations

2.9.1Solving Quadratics by Factorising 2.9.2Solving Quadratics Using the Formula

2.10

Simultaneous Equations

2.10.1Solving Simultaneous Equations (Elimination)

2.11

Iteration

2.11.1Iterative Formulae

2.12

Sequences

2.12.1Term-to-Term Rules 2.12.2Position-to-Term Rules 2.12.3Quadratic Sequence Formulae

2.13

Linear Graphs

2.13.1Equations of Straight Lines 2.13.2Finding Line Equation from Two Points 2.13.3Parallel Line Equations 2.13.4Perpendicular Line Equations

2.14

Introduction to Algebra

2.14.1Algebra Basics 2.14.2Substituting Values 2.14.3Algebraic Vocabulary 2.14.4Collecting Terms 2.14.5End of Topic Test - Algebra Basics & Power Rules 2.14.6Expanding Brackets I 2.14.7Common Factors 2.14.8Expanding Brackets II 2.14.9Rearrange For x 2.14.10Rearrange with Powers 2.14.11Exam-Style Questions - Basics of Algebra 2.14.12End of Topic Test - Expanding & Rearranging

2.15

Manipulating Algebra

2.15.1How to Factorise 2.15.2Factorising Quadratics 2.15.3Algebraic Fractions - Addition & Subtraction 2.15.4Algebraic Fractions - Multiplication & Division 2.15.5Diagnostic Misconceptions - Common Denominators 2.15.6Factorising Quadratics (tt)2.15.7Simplify algebraic fractions - cancelling factors 2.15.8Simplify algebraic fractions - factorisation

2.16

Proofs & Functions

2.16.1Identities & Proofs 2.16.2Exam-Style Questions - Algebraic Proof 2.16.3End of Topic Test - Algebra & Proofs 2.16.4Functions 2.16.5Combine Functions 2.16.6Composite Functions 2.16.7Inverse Functions 2.16.8Functions (tt)

2.17

Straight Line Graphs

2.17.1Straight Line Graphs 2.17.2y = mx + c 2.17.3Grad + YIntercept 2.17.4Exam-Style Questions - Straight Line Graphs 2.17.5End of Topic Test - Functions & Straight Lines 2.17.6Diagnostic Misconceptions - The x Coefficient

2.18

Common Graphs

2.18.1Shape of Quadratic Graphs 2.18.2Roots of a Quadratic Graph 2.18.3Completing Squares 2.18.4Turning Points 2.18.5Common Functions 1 2.18.6Common Functions 2 2.18.7Trigonometric functions 2.18.8End of Topic Test - Common Graphs

2.19

Transformations & Tangents

2.19.1Transformations 2.19.2Tangent on a Circle

2.20

Properties of Graphs

2.20.1Estimate Gradient 2.20.2Estimate Area 2.20.3Represent Gradient & Area 2.20.4Distance-Time Graph 2.20.5Velocity-Time Graph 2.20.6End of Topic Test - Transformations & Graphs

2.21

Solving Equations

2.21.1Solve Linear Equations 2.21.2Linear Equations with Fractions 2.21.3Solve Linear Equations Graphically 2.21.4Quadratic ±2.21.5Factorise Quadratics 2.21.6Quadratic Formula 2.21.7Complete the Square 2.21.8Solve Quadratics Graphically 2.21.9End of Topic Test - Solving Equations 2.21.10Simultaneous Equations 2.21.11Quadratic Simultaneous Equations 2.21.12Simultaneous Equations - Graphs 2.21.13Simultaneous Equations - Word Problems 2.21.14Approximate Solutions - Number of Iterations 2.21.15Approximate Solutions - Iterations to 1 DP 2.21.16Approximate Solutions - Iteration Formula 2.21.17Exam-Style Questions - Simultaneous Equations 2.21.18End of Topic Test - Simultaneous Equations 2.21.19Diagnostic Misconceptions - Solving Quadratics

2.22

Inequalities

2.22.1Solving Inequalities 2.22.2Solving Double Inequalities 2.22.3Solving Quadratic Inequalities 2.22.4Representing Inequalities 2.22.5Exam-Style Questions - Inequalities 2.22.6End of Topic Test - Inequalities

2.23

Sequences

2.23.1Nth Term Rules 2.23.2Recurrence Relationships 2.23.3Find Nth Term in Linear Sequence 2.23.4Check If a Number is In a Sequence 2.23.5Exam-Style Questions - Sequences 2.23.6Arithmetic & Geometric Progressions 2.23.7Key Sequences 2.23.8Quadratic Sequences 2.23.9Find Nth Term in Quadratic Sequence 2.23.10Exam-Style Questions - Geometric Series 2.23.11End of Topic Test - Sequences 2.23.12Nth Term Rules (tt)

Ratio

3.1

Unit Conversions

3.1.1Converting Units of Time 3.1.2Metric Conversions 3.1.3Converting Units of Area & Volume

3.2

Compound Units

3.2.1Calculating Speed 3.2.2Density, Mass & Volume

3.3

Ratio Fundamentals

3.3.1Ratios as Fractions & Linear Equations 3.3.2Sharing in a Ratio 3.3.3Proportion Problems

3.4

Advanced Ratio and Scale

3.4.1Map Scales & Real Distances

3.5

Proportion

3.5.1Currency Conversions 3.5.2Direct & Inverse Proportion

3.6

Algebraic Proportion

3.6.1Inverse Proportion Equations 3.6.2Direct Proportion with Powers & Roots 3.6.3Inverse Proportion with Powers & Roots

3.7

Ratios in Practice

3.7.1Converting Between Units 3.7.2Formula Triangles 3.7.3Scale Factors 3.7.4Ratios in Shapes 3.7.5Diagnostic Misconceptions - Writing Ratios

3.8

Manipulating Ratios

3.8.1Simplifying Ratios 3.8.2Turn Ratios into Equations 3.8.3Ratio Notation & Reduction 3.8.4End of Topic Test - Units & Ratios 3.8.5Part : Part Ratio 3.8.6Part : Whole Ratio 3.8.7Two Ratios 3.8.8Differences in Ratios 3.8.9Fractions in Ratios 3.8.10Numbers as Fractions 3.8.11Ratios to Fractions 3.8.12Exam-Style Questions - Ratios 3.8.13End of Topic Test - Fractions & Ratios 3.8.14Diagnostic Misconceptions - Combining Ratios

3.9

Percentage & Interest

3.9.1Convert Percentages to Decimals 3.9.2Convert Percentages to Fractions 3.9.3Find the Percentage 3.9.4More/Less Percentage 3.9.5Price from Percentage Change 3.9.6Calculate Percentage Change from Price 3.9.7End of Topic Test - Percentages 3.9.8Profit & Interest & Discount 3.9.9Original Price after Discounts 3.9.10Simple Interest 3.9.11Compound Growth & Decay 3.9.12Compound Interest 3.9.13Compound Interest - After How Many Days?3.9.14End of Topic Test - Percentages in Finance

3.10

Proportion

3.10.1Proportion 3.10.2Direct Proportion 3.10.3Inverse Proportion 3.10.4Exam-Style Questions - Proportionality

3.11

Gradient

3.11.1Gradient on a Straight Line 3.11.2Direct & Inverse Proportions as Graphs 3.11.3Gradient of a Curve 3.11.4Positive & Negative Gradients 3.11.5End of Topic Test - Proportion & Gradient

Geometry

4.1

Area & Perimeter

4.1.1Perimeter of Simple Shapes 4.1.2Perimeter of Compound Shapes 4.1.3Area of Triangles 4.1.4Area of Rectangles & Parallelograms 4.1.5Area of Trapezia 4.1.6Area of Compound Shapes 4.1.7Area of Compound Triangle Shapes

4.2

Angles

4.2.1Angles on a Line & Around a Point 4.2.2Combining Angle Facts in Triangles 4.2.3Angles on Parallel Lines 4.2.4Interior & Exterior Angles in Polygons

4.3

Trigonometry

4.3.1Sides in Right-Angled Triangles 4.3.2Angles in Right-Angled Triangles 4.3.3Exact Values of Trigonometric Ratios 4.3.4Trigonometry in 3D Shapes 4.3.5Using the Sine Rule 4.3.6Using the Cosine Rule 4.3.7Area of a Triangle Using Sine

4.4

Similarity & Congruence

4.4.1Length Scale Factors in Similar Shapes

4.5

Circle Geometry

4.5.1Calculating Circumference 4.5.2Calculating Area of a Circle

4.6

Circle Theorems

4.6.1Standard Circle Theorems 4.6.2Cyclic Quadrilaterals & Segment Theorems 4.6.3Chord & Tangent Properties

4.7

Pythagoras

4.7.1Pythagoras' Theorem in 2D 4.7.2Pythagoras’ Theorem in 3D

4.8

Surface Area

4.8.1Surface Area of Prisms 4.8.2Surface Area of Spheres & Cones

4.9

Volume

4.9.1Volume of Prisms 4.9.2Volume of Cylinders

4.10

Vectors

4.10.1Vectors & Column Notation 4.10.2Operations with Column Vectors

4.11

Introduction to Geometry

4.11.1Geometry Basics 4.11.2Geometry Basics 2 4.11.3Regular Polygons 4.11.4Triangles & Angles 4.11.5Midpoint 4.11.6Bisectors & Perpendiculars 4.11.7Locus 4.11.8Angle rules 4.11.9Parallel Angle Rules 4.11.10Angles in a Polygon 4.11.11Angles in a Triangle 4.11.12Exam-Style Questions - Angles in Shapes 4.11.13End of Topic Test - Geometry

4.12

Triangles & Quadrilaterals

4.12.1Types of Triangle 4.12.2Angles & Triangles 4.12.3Congruence 4.12.4Similarity 4.12.5Quadrilaterals

4.13

Transformations

4.13.1Types of Transformations 4.13.2Invariant Points 4.13.3Effects of Transformations 4.13.4Exam-Style Questions - Reflection

4.14

Circle Basics

4.14.1Properties of a Circle 1 4.14.2Chords 4.14.3Properties of a Circle 2 4.14.4Area of a Circle 4.14.5Circumference of a Circle 4.14.6Fractions of Circles 4.14.7Exam-Style Questions - Circle Formulae 4.14.8End of Topic Test - Triangles & Circles

4.15

Circle Theorems

4.15.1Perpendicular Bisector of a Chord 4.15.2Angle at the Centre 4.15.3Angle of a Semicircle 4.15.4Opposite Angles in Cyclic Quadrilaterals 4.15.5Tangent Lengths 4.15.6Angle in the Same Segment 4.15.7Alternate Segment Theorem 4.15.8Exam-Style Questions - Circle Theorems 4.15.9Diagnostic Misconceptions - Circumference & Radius

4.16

Measurements & Units

4.16.1Length 4.16.2Area 4.16.3Volume 4.16.4Capacity 4.16.5Time 4.16.6Money 4.16.7Mass 4.16.8Bearings 4.16.9Scale Factor & Enlargement 4.16.10Exam-Style Questions - Enlargements 4.16.11Diagnostic Misconceptions - Hours

4.17

Calculating Area

4.17.1Area of 2D Shapes 4.17.2Exam-Style Questions - Area 4.17.3End of Topic Test - Circle Theorems, Units & Area

4.18

Triangle Formulae

4.18.1Pythagoras' Theorem 4.18.2Pythagoras in 3D 4.18.3Trigonometry 4.18.4Exact Values 4.18.5Sine Rule 4.18.6Cosine Rule 4.18.7Area of a Triangle 4.18.8Exam-Style Questions - Trigonometry & Area 4.18.9Diagnostic Misconceptions - Bearings from Angles 4.18.10Diagnostic Misconceptions - Pythagoras' Theorem

4.19

3D Shapes

4.19.1Types of 3D Shape 4.19.2Plans & Elevations 4.19.3Nets 4.19.4Volume of 3D Shapes 4.19.5Volume of a Cuboid 4.19.6Volume of a Cylinder 4.19.7Volume of a Sphere 4.19.8Volume of a Hemisphere 4.19.9Volume of a Cone 4.19.10Volume of a Pyramid 4.19.11Surface Area & Composite Shapes 4.19.12Surface Area of a Cuboid 4.19.13Surface Area of a Cylinder 4.19.14Surface Area of a Sphere/Hemisphere 4.19.15Surface Area of a Cone 4.19.16Exam-Style Questions - Volume

4.20

Vectors

4.20.1Vectors Introduction 4.20.2Vectors 4.20.3Translation 4.20.4Scalar 4.20.5Exam-Style Questions - Vectors 4.20.6End of Topic Test - Trigonometry, Shapes & Vectors 4.20.7Diagnostic Misconceptions - Column Vectors

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness 5.1.2Sum of Probabilities 5.1.3Relative & Expected Frequency 5.1.4Frequency Tables 5.1.5Using Frequency Trees 5.1.6Complete Frequency Trees

5.2

Representing Probabilities

5.2.1Probability 5.2.2Probability Scale 5.2.3Sample Space Diagrams 5.2.4Sets 5.2.5Venn Diagrams 5.2.6Probabilities in Venn Diagrams 5.2.7Exam-Style Questions - Venn Diagrams

5.3

Probability Calculations

5.3.1Theoretical Probabilities 5.3.2Multiple Events 5.3.3Probability Calculations 5.3.4Dependent Events 5.3.5Independent Events 5.3.6Probability Tree Diagram 5.3.7Complex Tree Diagrams 5.3.8Conditional Probability 5.3.9Conditional Probability From a Two Way Table 5.3.10Conditional Probability Venn Diagrams 5.3.11Conditional Probability Tree Diagrams 5.3.12Exam-Style Questions - Probability 5.3.13End of Topic Test - Probability 5.3.14Diagnostic Misconceptions - Denominator

Statistics

Jump to other topics

Number

1.1

Place Value & Ordering

1.1.1Place Value

1.2

Numerical Operations

1.3

Structure of Calculations

1.3.1Order of Operations (BIDMAS)

1.4

Fractions

1.4.1Writing Fractions 1.4.2Simplifying Fractions 1.4.3Fractions & Recurring Decimals

1.5

Numerical Operations on Fractions

1.6

Roots & Powers

1.6.1Roots 1.6.2Fractional Indices (1/n)1.6.3Fractional Indices (m/n)1.6.4Laws of Indices (Multiplication)1.6.5Laws of Indices (Division)

1.7

Numerical Skills

1.7.1Using a Calculator - Powers, Roots & Fractions 1.7.2Using a Calculator - Trigonometric Functions

1.8

Factors & Multiples

1.8.1Finding Factors 1.8.2Finding Multiples

1.9

Surds

1.9.1Multiplying & Dividing Surds 1.9.2Simplifying Surds 1.9.3Expanding Expressions With Surds 1.9.4Rationalising the Denominator

1.10

Standard Form

1.11

Fractions, Decimals & Percentages

1.11.1Converting Fractions, Decimals & Percentages 1.11.2Converting Recurring Decimals 1.11.3Numbers as Percentages 1.11.4Fractions of Amounts

1.12

Percentage Change

1.12.1Percentages of Amounts (No Calculator)1.12.2Percentage Change 1.12.3Original Values After Percentage Change 1.12.4Growth & Decay Using Multipliers

1.13

Rounding & Estimation

1.13.1Rounding Integers 1.13.2Estimating Calculations 1.13.3Rounding Decimals 1.13.4Rounding With Significant Figures

1.14

Bounds & Accuracy

1.14.1Finding Error Intervals 1.14.2Upper & Lower Bounds 1.14.3Error Intervals From Truncation

1.15

Using Numbers

1.16

Fractions, Decimals & Percentages

1.17

Powers & Roots

1.18

Accuracy

Algebra

2.1

Algebraic Notation

2.1.1Using Algebraic Notation

2.2

Substitution & Simplification

2.2.1Substituting into Expressions 2.2.2Substituting into Real-Life Formulae 2.2.3Simplifying Expressions by Collecting Like Terms

2.3

Brackets

2.4

Algebraic Fractions

2.4.1Simplifying Algebraic Fractions by Cancelling 2.4.2Simplifying Algebraic Fractions by Factorising 2.4.3Operations with Algebraic Fractions

2.5

Rearranging

2.5.1Changing the Subject 2.5.2Changing the Subject (Repeated Subject)

2.6

Functions

2.6.1Finding the Input of a Function 2.6.2Inverse Functions

2.7

Solving Linear Equations

2.8

Inequalities

2.8.1Solving Linear Inequalities 2.8.2Linear Inequalities with Unknowns on Both Sides

2.9

Solving Quadratic Equations

2.9.1Solving Quadratics by Factorising 2.9.2Solving Quadratics Using the Formula

2.10

Simultaneous Equations

2.10.1Solving Simultaneous Equations (Elimination)

2.11

Iteration

2.11.1Iterative Formulae

2.12

Sequences

2.12.1Term-to-Term Rules 2.12.2Position-to-Term Rules 2.12.3Quadratic Sequence Formulae

2.13

Linear Graphs

2.13.1Equations of Straight Lines 2.13.2Finding Line Equation from Two Points 2.13.3Parallel Line Equations 2.13.4Perpendicular Line Equations

2.14

Introduction to Algebra

2.15

Manipulating Algebra

2.16

Proofs & Functions

2.17

Straight Line Graphs

2.18

Common Graphs

2.19

Transformations & Tangents

2.19.1Transformations 2.19.2Tangent on a Circle

2.20

Properties of Graphs

2.20.1Estimate Gradient 2.20.2Estimate Area 2.20.3Represent Gradient & Area 2.20.4Distance-Time Graph 2.20.5Velocity-Time Graph 2.20.6End of Topic Test - Transformations & Graphs

2.21

Solving Equations

2.22

Inequalities

2.23

Sequences

Ratio

3.1

Unit Conversions

3.1.1Converting Units of Time 3.1.2Metric Conversions 3.1.3Converting Units of Area & Volume

3.2

Compound Units

3.2.1Calculating Speed 3.2.2Density, Mass & Volume

3.3

Ratio Fundamentals

3.3.1Ratios as Fractions & Linear Equations 3.3.2Sharing in a Ratio 3.3.3Proportion Problems

3.4

Advanced Ratio and Scale

3.4.1Map Scales & Real Distances

3.5

Proportion

3.5.1Currency Conversions 3.5.2Direct & Inverse Proportion

3.6

Algebraic Proportion

3.6.1Inverse Proportion Equations 3.6.2Direct Proportion with Powers & Roots 3.6.3Inverse Proportion with Powers & Roots

3.7

Ratios in Practice

3.7.1Converting Between Units 3.7.2Formula Triangles 3.7.3Scale Factors 3.7.4Ratios in Shapes 3.7.5Diagnostic Misconceptions - Writing Ratios

3.8

Manipulating Ratios

3.9

Percentage & Interest

3.10

Proportion

3.10.1Proportion 3.10.2Direct Proportion 3.10.3Inverse Proportion 3.10.4Exam-Style Questions - Proportionality

3.11

Gradient

3.11.1Gradient on a Straight Line 3.11.2Direct & Inverse Proportions as Graphs 3.11.3Gradient of a Curve 3.11.4Positive & Negative Gradients 3.11.5End of Topic Test - Proportion & Gradient

Geometry

4.1

Area & Perimeter

4.2

Angles

4.2.1Angles on a Line & Around a Point 4.2.2Combining Angle Facts in Triangles 4.2.3Angles on Parallel Lines 4.2.4Interior & Exterior Angles in Polygons

4.3

Trigonometry

4.4

Similarity & Congruence

4.4.1Length Scale Factors in Similar Shapes

4.5

Circle Geometry

4.5.1Calculating Circumference 4.5.2Calculating Area of a Circle

4.6

Circle Theorems

4.6.1Standard Circle Theorems 4.6.2Cyclic Quadrilaterals & Segment Theorems 4.6.3Chord & Tangent Properties

4.7

Pythagoras

4.7.1Pythagoras' Theorem in 2D 4.7.2Pythagoras’ Theorem in 3D

4.8

Surface Area

4.8.1Surface Area of Prisms 4.8.2Surface Area of Spheres & Cones

4.9

Volume

4.9.1Volume of Prisms 4.9.2Volume of Cylinders

4.10

Vectors

4.10.1Vectors & Column Notation 4.10.2Operations with Column Vectors

4.11

Introduction to Geometry

4.12

Triangles & Quadrilaterals

4.12.1Types of Triangle 4.12.2Angles & Triangles 4.12.3Congruence 4.12.4Similarity 4.12.5Quadrilaterals

4.13

Transformations

4.13.1Types of Transformations 4.13.2Invariant Points 4.13.3Effects of Transformations 4.13.4Exam-Style Questions - Reflection

4.14

Circle Basics

4.15

Circle Theorems

4.16

Measurements & Units

4.17

Calculating Area

4.17.1Area of 2D Shapes 4.17.2Exam-Style Questions - Area 4.17.3End of Topic Test - Circle Theorems, Units & Area

4.18

Triangle Formulae

4.19

3D Shapes

4.20

Vectors

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness 5.1.2Sum of Probabilities 5.1.3Relative & Expected Frequency 5.1.4Frequency Tables 5.1.5Using Frequency Trees 5.1.6Complete Frequency Trees

5.2

Representing Probabilities

5.2.1Probability 5.2.2Probability Scale 5.2.3Sample Space Diagrams 5.2.4Sets 5.2.5Venn Diagrams 5.2.6Probabilities in Venn Diagrams 5.2.7Exam-Style Questions - Venn Diagrams

5.3

Probability Calculations

Statistics

Practice questions on Rationalise Denominator

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1
Which of these are in exact form?True / false
2
Which of these are in exact form?True / false
3
Which of these are in exact form?Multiple choice
4
How should you rationalise the denominator of the fraction: $$\frac{8}{\sqrt{7}}$$?Multiple choice

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Simplify Surds

Exam-Style Questions - Indices