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Glossary
Chapter 1: Number skills
- Overview
- 1.1 Place value and ordering numbers
- 1.2 The four operations
- 1.3 Negative numbers
- 1.4 Priority of operations and inverse operations
- 1.5 Rounding and estimation
- Review
Chapter 2: Factors, multiples and primes
- Overview
- 2.1 Factors, multiples and prime numbers
- 2.2 Prime factorisation, HCF and LCM
- 2.3 Systematic listing strategies
- Review
Chapter 3: Powers, roots and indices
- Overview
- 3.1 Powers and roots
- 3.2 Estimating powers and roots
- 3.3 Laws of indices
- Review
Chapter 4: Fractions, decimals and percentages
- Overview
- 4.1 Equivalent fractions and ordering
- 4.2 Adding and subtracting fractions
- 4.3 Multiplying and dividing fractions
- 4.4 Fractions and terminating decimals
- 4.6 Fractions and percentages as operators
- Review
Chapter 5: Ratio and proportion
- Overview
- 5.1 Ratio notation, simplifying and sharing
- 5.2 Ratio, fractions and linear functions
- 5.3 Direct proportion
- 5.4 Inverse proportion
- 5.5 Scale factors, scale diagrams and maps
- Review
Chapter 6: Percentages and financial mathematics
- Overview
- 6.1 Percentage of an amount
- 6.2 Percentage change
- 6.3 Reverse percentages
- 6.4 Simple and compound interest
- 6.5 Growth, decay and repeated percentage change
- Review
Chapter 7: Measures, accuracy, standard form and surds
- Overview
- 7.1 Units of measure and conversions
- 7.2 Compound measures: speed, density and pressure
- 7.3 Rounding and error intervals
- 7.5 Standard form
- Review
Chapter 8: Algebraic expressions and manipulation
- Overview
- 8.1 Algebraic notation and vocabulary
- 8.2 Simplifying and collecting like terms
- 8.3 Expanding brackets
- 8.4 Factorising with common factors
- 8.5 Substitution into expressions and formulae
- 8.6 Rearranging formulae
- Review
Chapter 9: Equations and inequalities
- Overview
- 9.1 Solving linear equations
- 9.2 Equations with unknowns on both sides
- 9.3 Forming and solving equations
- 9.4 Inequalities on a number line and set notation
- 9.5 Solving linear inequalities
- Review
Chapter 10: Sequences
- Overview
- 10.1 Term-to-term and position-to-term rules
- 10.2 Special sequences
- 10.3 The nth term of a linear sequence
- Review
Chapter 11: Straight-line graphs and rates of change
- Overview
- 11.1 Coordinates and plotting graphs
- 11.2 Gradient and intercept: y = mx + c
- 11.3 Finding the equation of a line
- 11.5 Real-life and distance-time graphs
- Review
Chapter 12: Quadratic expressions, equations and graphs
- Overview
- 12.1 Expanding and factorising quadratics
- 12.3 Quadratic graphs and their features
- 12.4 Solving quadratic equations by factorising
- Review
Chapter 13: Other functions and graphs
- Overview
- Review
Chapter 14: Simultaneous equations and iteration
- Overview
- 14.1 Simultaneous equations by elimination
- 14.2 Simultaneous equations by substitution
- 14.3 Solving simultaneous equations graphically
- Review
Chapter 15: Functions, algebraic fractions and proof
- Overview
- 15.4 Identities and equivalent expressions
- Review
Chapter 16: Angles, polygons and geometrical reasoning
- Overview
- 16.1 Angle facts at a point and on a line
- 16.2 Angles in parallel lines
- 16.3 Angles in triangles and quadrilaterals
- 16.4 Angles in polygons
- Review
Chapter 17: Properties, constructions, loci and bearings
- Overview
- 17.1 Properties of 2D and 3D shapes
- 17.2 Plans and elevations
- 17.3 Constructions with ruler and compasses
- 17.4 Loci
- 17.5 Bearings and scale drawings
- Review
Chapter 18: Transformations, congruence and similarity
- Overview
- 18.1 Reflection, rotation and translation
- 18.2 Enlargement
- 18.4 Congruence and congruence criteria
- 18.5 Similarity and similar lengths
- Review
Chapter 19: Perimeter, area, surface area and volume
- Overview
- 19.1 Perimeter and area of 2D shapes
- 19.2 Circumference and area of circles
- 19.3 Arc length and sector area
- 19.4 Surface area and volume of prisms and cylinders
- 19.5 Surface area and volume of spheres, cones and pyramids
- 19.6 Composite shapes and solids
- Review
Chapter 20: Pythagoras' theorem and trigonometry
- Overview
- 20.1 Pythagoras' theorem
- 20.2 Trigonometric ratios (SOHCAHTOA)
- 20.3 Angles of elevation and depression
- 20.4 Exact trigonometric values
- Review
Chapter 21: Circle geometry and circle theorems
- Overview
- 21.1 Parts of a circle
- Review
Chapter 22: Vectors
- Overview
- 22.1 Vector notation and column vectors
- 22.2 Adding and subtracting vectors
- Review
Chapter 23: Probability
- Overview
- 23.1 The probability scale and equally likely outcomes
- 23.2 Relative frequency and expected outcomes
- 23.3 Frequency trees, tables and possibility spaces
- 23.4 Combined events and tree diagrams
- 23.5 Venn diagrams and set notation
- Review
Chapter 24: Statistics: representing and analysing data
- Overview
- 24.1 Collecting data, sampling and questionnaires
- 24.2 Tables, bar charts, pie charts and pictograms
- 24.3 Averages and range
- 24.4 Averages from frequency tables
- 24.5 Time series and line graphs
- Review
Chapter 25: Statistics: spread, grouped data and correlation
- Overview
- 25.4 Scatter graphs and correlation
- Review
About this book

3.2 Estimating powers and roots — Practise with help

Warm-up: remember

Work out $8^2$ .
Work out $\sqrt{81}$ .

Answers and feedback

$64$ , because $8\times8=64$ .
$9$ , because $9^2=81$ .

Practise with help

1. Estimate $\sqrt{45}$ to the nearest whole number.

Hint

Find the two consecutive square numbers around 45.

One step

$6^2=36$ and $7^2=49$ . The midpoint is $42.5$ .

Full working

$45$ is between $36$ and $49$ , and is above their midpoint, $42.5$ . Therefore $\sqrt{45}$ rounds to $7$ .

2. Estimate $\sqrt[3]{300}$ to the nearest whole number.

Hint

Compare 300 with $6^3$ and $7^3$ .

One step

$6^3=216$ and $7^3=343$ . The midpoint is $279.5$ .

Full working

$300$ is above the midpoint of $216$ and $343$ . Therefore $\sqrt[3]{300}$ rounds to $7$ .

3. Estimate $31^2$ by rounding 31 to the nearest ten.

Hint

Round 31 first, then square the rounded number.

One step

$31\approx30$ .

Full working

$31^2\approx30^2=900$ .