10.2 Exponents and Indices

Revision slides, flash cards, quizzes, revision notes and a text version of the slides for this unit. Open to everyone, no password needed.

Slide 1 of 10

10.2 revision slides as a PDF

Flash cards

Click a card to flip it. Say the answer first, then check.

Print the cards (PDF): word on the front, answer on the back, double-sided, flip on the long edge

Quizzes

Pick an answer to see if you are right. Your score shows at the end of each quiz.

Revision notes

Download and print the revision notes (PDF, A4)

The big idea. Why keep an answer exact when a calculator gives a decimal? Index laws, negative and fractional indices, exponential equations, surds and rationalising denominators.

Key vocabulary

Word	Meaning
Index (power, exponent)	the small number that says how many times to multiply the base, e.g. the 3 in 2³
Base	the number that is being multiplied by itself, e.g. the 2 in 2³
Negative index	an index that gives a reciprocal: 2⁻³ = 1/2³ = 1/8
Fractional index	an index that means a root: 8^(1/3) is the cube root of 8
Reciprocal	1 divided by a number: the reciprocal of 8 is 1/8
Exponential equation	an equation with the unknown in the index, e.g. 2^x = 32
Rational number	a number that can be written as a fraction of two integers
Irrational number	a number that cannot be written as a fraction, e.g. √2
Surd	an irrational root left in exact form, such as √2 or √12
Like surds	surds with the same root part, e.g. 3√2 and 5√2, which can be added or subtracted
Numerator	the number or expression above the line in a fraction
Denominator	the number or expression below the line in a fraction
Rationalise the denominator	rewrite a fraction so that the denominator has no surd
Conjugate	the same two terms with the opposite sign between them: the conjugate of 3 + √2 is 3 − √2

Index laws (positive and zero indices)

a^m × aⁿ = a^m+n (same base: add the powers). a^m ÷ aⁿ = a^m−n.
(a^m)ⁿ = a^mn. (ab)ⁿ = aⁿbⁿ.
a⁰ = 1 because aⁿ ÷ aⁿ = a⁰ = 1.
Expand to see why: x³ × x² = (x·x·x)(x·x) = x⁵.
The laws need the same base: 2³ × 3² cannot be combined.

Negative and fractional indices

a⁻ⁿ = 1 ÷ aⁿ: a negative index is a reciprocal. 2⁻³ = 1/8.
(a/b)⁻ⁿ = (b/a)ⁿ: flip, then use the positive power.
a^1/n = ⁿ√a. 9^1/2 = 3, 27^1/3 = 3.
a^m/n = (ⁿ√a)^m. 8^2/3 = (³√8)² = 4.
The pattern 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2 shows why.

Exponential equations

If a^x = a^y (same base, a > 0, a ≠ 1) then x = y.
Write both sides with the same base: 2^x = 32 = 2⁵, so x = 5. 9^x = 27 gives 3^2x = 3³ and x = 3/2.
Answers can be negative or fractional: 2^x = 1/8 = 2⁻³ gives x = −3.
With an algebraic exponent: 4^x+1 = 2^x+7 → 2^2x+2 = 2^x+7 → 2x + 2 = x + 7 → x = 5. Check by substituting.
The rule fails for base 1 and 0.

Surds

A surd is an irrational root left exact, such as √2 or √12.
Simplify using the largest square factor: √12 = √4 × √3 = 2√3. √50 = 5√2.
Add/subtract like surds only: 3√2 + 5√2 = 8√2. √2 + √3 stays as it is.
Multiply: √a × √b = √(ab), and √a × √a = a. 2√3 × 3√2 = 6√6.
Expand brackets as normal: √2(3 + √2) = 3√2 + 2. (1 + √3)² = 1 + 2√3 + 3 = 4 + 2√3.

Rationalising the denominator

Rationalise = remove the surd from the denominator, because a rational denominator is easier to use and compare.
Single surd: 1/√2 = (1 × √2)/(√2 × √2) = √2/2. Multiplying by √2/√2 is multiplying by 1, so the value is unchanged.
Conjugate: the same terms with the opposite sign. The conjugate of 3 + √2 is 3 − √2.
1/(3 + √2) = (3 − √2)/((3 + √2)(3 − √2)) = (3 − √2)/(9 − 2) = (3 − √2)/7.
The product of a number and its conjugate is rational: (a + √b)(a − √b) = a² − b.

Sequences with surd terms

Simplify every surd term first: √2, √8, √18 = √2, 2√2, 3√2.
That is an arithmetic sequence with d = √2.
Geometric check: √2, 2, 2√2 has r = √2 (2 ÷ √2 = √2, and 2√2 ÷ 2 = √2).
Write u_n in exact form: u_n = √2 × n for the first sequence.

Common mistakes

✕ “x³ × x² = x⁶.” ✓ Add the powers: x⁵.
✕ “2⁻³ = −8.” ✓ A negative index gives a reciprocal: 1/8.
✕ “8^1/3 = 8/3.” ✓ A power of 1/3 is a cube root: 2.
✕ “√2 + √3 = √5.” ✓ Unlike surds cannot be added: √2 + √3 stays as it is.
✕ “(a + b)² = a² + b².” ✓ Expand: a² + 2ab + b². So (1 + √3)² = 4 + 2√3.

Practice booklet

Unit 10.2 Practice Booklet

mathscafe.net
Lorna Brown © 2026

Ms B’s mathscafe

Practice Booklet

MYP 4 · Year 10 Mathematics (Extended)

Unit 10.2 · Exponents and Indices

Name:

10.2.1: Index laws · Questions 1–20
10.2.1b: Negative and fractional indices · Questions 21–38
10.2.2: Comparing exponents · Questions 39–57
10.2.3: Algebraic exponents · Questions 58–76
10.2.4 (a): Surds: what they are and simplifying · Questions 77–95
10.2.4 (b): Multiplying and collecting surds · Questions 96–114
10.2.4b: Rationalising the denominator · Questions 115–132
10.2.5: Sequences with surds · Questions 133–151
Homework: Homework page and checking record
Answers: For checking and correcting in green pen

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice BookletMs B’s mathscafe · page 1

How to use this booklet and your homework

Read this first

Choosing your level

Every part of the unit has four levels. Start where you feel comfortable and move between them freely. Most questions ask you to explain as well as answer, because being able to say why is what makes a method stick.

Learning

The standard case with small numbers, with the method set out for you.

Practising

The method on its own, with your reasoning written out.

Challenging

Harder, with several steps, and finding what is wrong with someone else's working.

Mastery

Explaining why a rule must be true, and testing it on new cases.

Working on your homework

Pens and setting out

Answers in blue or black non-erasable pen, or in pencil.
Colour and highlighters are for annotating your work and showing the steps of a method.
Never erase a mistake. Rework it underneath so you can see what changed.

Green pen is for checking. When you finish, use the answers at the back to check your work.

Got it wrong? Circle the question number, then make your correction and write your error analysis in green pen.
Still can’t work out how to solve it? Put a star ★ next to the question number, so we know to go over it in class.

Each sheet

Your name goes on every sheet, because they are handed in one at a time.
Each question is worth up to 3 marks. If a question has parts (a, b, c…), each part is marked out of 3. At the end of each level, write your score out of the total for that level.
Questions are numbered 1, 2, 3… all the way through the booklet, so we can say exactly which ones to do.
If you are stuck after trying, ask for help. Showing me what you tried is a good start.

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 2

10.2.1 Index laws

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
base	the number being multiplied by itself	2 in 2³
index	how many times the base is used as a factor	3 in 2³
power	a base with an index	2³ = 8
same base	the same number or letter	5² and 5⁴

Laws (same base only): a^m × aⁿ = a^m+n · a^m ÷ aⁿ = a^m−n · (a^m)ⁿ = a^mn · a⁰ = 1

Learning

Write 2 × 2 × 2 × 2 as a power

a) Write 2³ × 2² in full, with every 2 showing b) Count the 2s and write the answer as one power of 2

Write 3⁴ × 3⁵ as a single power of 3

Write 7⁸ ÷ 7³ as a single power of 7

Write (5²)³ in full, then as a single power of 5

Work out a) 9⁰ b) 100⁰ c) 1⁰

Learning score / 27

Practising

Simplify a) x⁴ × x³ b) y⁹ ÷ y² c) (a³)⁵

Simplify (2x³)⁴

Simplify 12x⁵3x²

Simplify 3a²b × 4ab³

Simplify (3a²b)³

Practising score / 21

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 3

10.2.1 Index laws

Challenging · Mastery

Name:

Challenging

Simplify (2x³)² × 3x⁴6x⁵

Work out 4⁰ + (3x)⁰ + 5x⁰, where x is not zero

Find n: a) 3ⁿ ÷ 3⁴ = 3⁶ b) (5ⁿ)² = 5¹²

Find and describe the mistake. Sam writes 2³ × 3² = 6⁵

Write 2³ × 2³ × 2³ as a power of 2 in two different ways

Challenging score / 18

Mastery

Use the divide law on 5³ ÷ 5³ to explain why 5⁰ = 1

Explain why 3⁴ × 3² = 3⁶, but 3⁴ + 3² is not 3⁶. Work out both sides to check

Show that (2³)⁴ = (2⁴)³, then explain why (a^m)ⁿ = (aⁿ)^m

A square has side 2⁵ cm. Write its area as a power of 2 and work it out

Mastery score / 12

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 4

10.2.1b Negative and fractional indices

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
reciprocal	a fraction turned upside down	the reciprocal of 4 is 14
negative index	means take the reciprocal	a⁻ⁿ = 1aⁿ
fractional index	means a root; the denominator is the root	a^mn = (n-th root of a)^m
root	undoes a power	the cube root of 8 is 2

Do the root first, then the power.

Learning

Continue the pattern. 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = ___, 2⁻² = ___

Write as a fraction a) 2⁻³ b) 10⁻² c) 5⁻¹

Work out a) 25¹² b) 100¹² c) 8¹³ d) 27¹³

Write the reciprocal of a) 4 b) 23

Learning score / 30

Practising

Evaluate a) 3⁻² b) 4⁻³ c) 10⁻⁴

Evaluate a) (12)⁻⁴ b) (23)⁻²

Evaluate a) 16¹⁴ b) 32¹⁵ c) 1000¹³

Evaluate a) 8²³ b) 27²³ c) 4³²

Write 3⁻² × 3⁵ as a single power of 3, then evaluate it

Practising score / 36

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 5

10.2.1b Negative and fractional indices

Challenging · Mastery

Name:

Challenging

Evaluate a) 9⁻¹² b) 27⁻²³ c) 16⁻³⁴

Evaluate (49)⁻³²

Simplify x⁻² × x⁵ ÷ x⁻¹

Find and describe the mistake. Jo says 2⁻³ = −8

a) Write 132 as a power of 2 b) Write 8 × 2⁻⁵ as a single power of 2, then as a fraction

Challenging score / 24

Mastery

Use a¹² × a¹² to explain why a¹² must be the square root of a

A student says 4³² = 6 because 4 × 3 ÷ 2 = 6. Explain the mistake and give the correct value

Evaluate 8⁻²³, saying which rule you use at each step

Put in order, smallest first, without a calculator: 2⁻¹, 4¹², 8⁰, 27⁻¹³

Mastery score / 12

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 6

10.2.2 Comparing exponents

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
common base	the same base on both sides	4 and 8 are both powers of 2
equate	make two things equal	2^x = 2⁵, so x = 5

Steps: 1 choose a common base. 2 rewrite each side as a power of it. 3 equate the exponents. 4 solve and check.

Learning

Solve 2^x = 8

Solve a) 5^x = 25 b) 3^x = 81 c) 10^x = 1000

Write as a power of 2 a) 4 b) 16 c) 32

Solve 2^x = 64 by writing 64 as a power of 2

Solve 5^x = 1

Learning score / 27

Practising

Solve a) 2^x = 256 b) 5^x = 625 c) 10^x = 1 000 000

Solve 25^x = 125

Solve 4^x = 116

Solve 8^x = 16

Solve 9^x = 27

Practising score / 21

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 7

10.2.2 Comparing exponents

Challenging · Mastery

Name:

Challenging

Solve a) 27^x = 9 b) 3^x = 181 c) 8^x = 14

Solve 4^x = 32

Solve (12)^x = 8

Find and describe the mistake. Dee solves 9^x = 27. She writes 3^x = 3³, so x = 3

Solve 2^x = 0.25

Challenging score / 21

Mastery

Explain why 3^x = 5 cannot be solved by writing both sides as a power of 3. Between which two whole numbers is x?

Explain why 1^x = 1 has many solutions, so we cannot compare exponents when the base is 1

a) A culture of bacteria starts with 1 cell and doubles every hour. After how many hours are there 1024 cells? b) Another culture triples every hour. After how many hours does it reach 729 cells?

Solve 16^x = 8 and 8^y = 16. Show that x × y = 1 and explain why

Mastery score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 8

10.2.3 Algebraic exponents

Learning · Practising

Name:

Key vocabulary and steps

Word	Meaning
exponent	the power; here it contains x
common base	write both sides as powers of the same base
equate	make the exponents equal, then solve

Steps: 1 match the bases. 2 use the index laws on the exponents. 3 equate the exponents and solve. 4 check by substitution.

Learning

Write 8 as a power of 2

Simplify a) 2³ × 2⁴ b) (2³)²

Write as a power of 2 a) 4^x b) 8^x c) 16^x

Solve 2^x+1 = 2⁵

Solve 3^2x = 3⁸

Learning score / 24

Practising

Solve 9^x = 3^x+4

Solve 5^x+1 = 25^x

Solve 2^x−1 = 4^x+1

Solve 3^2x+1 = 27^x−1

Solve 5^x+2 = 25^x−3

Practising score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 9

10.2.3 Algebraic exponents

Challenging · Mastery

Name:

Challenging

Solve 2^3x−1 = 4^x+2

Solve 4^x+1 = 8^x

Solve (12)^x = 4^x+3

Find and describe the mistake. Ali solves 4^x+1 = 2^x+5. He writes 4^x+1 = 2^2x+1

Check by substitution that x = 3 is a solution of 4^x+1 = 2^x+5

Challenging score / 15

Mastery

Solve 2^x × 4^x+1 = 128

Explain why 27^x−1 = 3^3x−3 and not 3^3x−1

a) A population is 5 × 2^t after t hours. When is it 160? b) When is 5 × 2^t equal to 5 × 4^t−2?

Write an equation of the form 3^ax+b = 9^cx+d whose solution is x = 2. Show that it works

Mastery score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 10

10.2.4 (a) Surds: what they are and simplifying

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
surd	a root that cannot be written exactly as a whole number or fraction	√7
irrational	a number whose decimal never ends or repeats	√2
perfect square	a whole number squared	1, 4, 9, 16, 25, 36, 49
highest perfect square factor	the largest square that divides the number	for 72 it is 36
simplest form	the number under the root has no square factor	6√2

Rule: √ab = √a × √b. Pull out the highest perfect square factor.

Learning

List the first ten perfect squares

Which of these are surds? √16 √7 √25 √10

Complete: √18 = √9 × √2 = ___ √2

Simplify √8 by writing it as √4 × √2

Simplify √12. Hint: which perfect square goes into 12?

Learning score / 15

Practising

Simplify √48

Simplify √75

Simplify √72, using the highest perfect square factor

Simplify √45

Simplify 5√8

Practising score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 11

10.2.4 (a) Surds: what they are and simplifying

Challenging · Mastery

Name:

Challenging

Simplify a) √180 b) 3√50 c) √200

Write 6√2 as a single surd √n

Find and describe the mistake. A student writes √9 + √16 = √25 = 5

Simplify √98 and √32

Simplify √755

Challenging score / 21

Mastery

A student simplifies √72 as 2√18. Is the value right? Is it in simplest form? Explain

Show that √8 and √18 are both whole-number multiples of √2

Explain why √2 cannot be written exactly as a fraction or a finite decimal

A square has area 50 cm². Find the exact side length in simplest form

Mastery score / 12

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 12

10.2.4 (b) Multiplying and collecting surds

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
like surds	surds with the same number under the root	3√2 and 5√2
unlike surds	different numbers under the root, so they cannot be collected	√2 and √3

Rules: √a × √b = √ab · √a × √a = a · collect like surds · expand with the box method, then simplify

Learning

Work out √2 × √3

Work out √5 × √5

Work out 3√2 × 2√5

Simplify 4√3 + 2√3

100

Simplify 7√5 − 3√5

Learning score / 15

Practising

101

Simplify 3√2 × 4√6

102

Expand √3(5 − √3)

103

Expand (3 + √2)(1 + √2)

104

Simplify √48 − √12

105

Simplify √8 + √50

Practising score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 13

10.2.4 (b) Multiplying and collecting surds

Challenging · Mastery

Name:

Challenging

106

Expand (3 + √2)²

107

Expand (4 + √3)(4 − √3)

108

Expand (√5 + 2)(√5 − 2)

109

Find and describe the mistake. Tom writes (3 + √2)² = 9 + 2 = 11

110

Simplify √45 + √20 − √5

Challenging score / 15

Mastery

111

A student writes √3 + √2 = √5. Use a calculator to show this is wrong, and explain why

112

Find the exact area of a square with side (2 + √3) cm

113

A right-angled triangle has shorter sides 2√3 cm and √6 cm. Find the hypotenuse exactly

114

Find x if √x + √4x = 3√5

Mastery score / 12

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 14

10.2.4b Rationalising the denominator

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
rationalise	rewrite so the denominator is a whole number	1√2 = √22
conjugate	same terms, opposite sign between them	3 + √2 and 3 − √2
difference of two squares	(a + b)(a − b) = a² − b²	(3 + √2)(3 − √2) = 7

Multiply top and bottom by √b, or by the conjugate of the denominator.

Learning

115

Work out √7 × √7

116

Work out 1√2 × √2√2 and write the answer as a fraction

117

Rationalise a) 3√3 b) 10√5

118

Write the conjugate of a) 3 + √2 b) 4 − √5 c) √7 + 1

Learning score / 21

Practising

119

Rationalise a) 6√3 b) 4√2 c) 15√5

120

Rationalise 32√3

121

Rationalise 103√2

122

Expand (4 + √3)(4 − √3)

123

Rationalise 53 + √2

Practising score / 21

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 15

10.2.4b Rationalising the denominator

Challenging · Mastery

Name:

Challenging

124

Rationalise 1√2 + 1

125

Rationalise 23 − √5

126

Rationalise 6√7 + 1

127

Find and describe the mistake. Ben rationalises 12 + √3 by multiplying the top and bottom by 2 + √3

128

Rationalise 4√5 − 1

Challenging score / 15

Mastery

129

Explain why multiplying a fraction by its conjugate over itself does not change its value, even though it looks different

130

Rationalise √5√5 + √2

131

Simplify 12 − √3 + 12 + √3

132

A rectangle has area 6 cm² and length (3 + √3) cm. Find the width in the form a − √b

Mastery score / 12

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 16

10.2.5 Sequences with surds

Learning · Practising

Name:

Key vocabulary

Word	Meaning	Example
common difference	d: the gap between terms in an arithmetic sequence	√3, 2√3, 3√3: d = √3
common ratio	r: what you multiply by each time in a geometric sequence	1, √5, 5: r = √5
general term	u_n: the rule for any term	u_n = n√3

Arithmetic: u_n = u₁ + (n − 1)d. Geometric: u_n = u₁ × rⁿ⁻¹. Simplify every surd first.

Learning

133

Simplify a) √12 b) √27

134

Simplify √20, √45 and √80

135

Write the next term: √2, 2√2, 3√2, …

136

a) Simplify each term of √3, √12, √27 b) Find the common difference d

137

Write u_n for √3, 2√3, 3√3, …

Learning score / 21

Practising

138

For √8, √32, √72: simplify each term, then find d and u_n

139

The sequence 2, 2√3, 6, 6√3 is geometric. Find r and the next term

140

For the geometric sequence 1, √5, 5, 5√5, … find r and u₆

141

The sequence √5, 3√5, 5√5, … is arithmetic. Find d, u_n and u₁₀

142

Is √2, 2, 2√2, 4 arithmetic or geometric? Give d or r

Practising score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

10.2.5 Sequences with surds

Challenging · Mastery

Name:

Challenging

143

Find u_n and u₂₀ for √2, √8, √18, …

144

A geometric sequence has u₁ = 3 and u₂ = 3√2. Find r, u₃ and u₅

145

An arithmetic sequence has u₁ = 2√3 and d = √3. Find u_n and show that u₈ = √243

146

Find and describe the mistake. Maya says √3, √12, √27 is geometric with r = √4 = 2, because 12 ÷ 3 = 4

147

Show that the ratio of 4, 2√2, 2, … is √22

Challenging score / 15

Mastery

148

Explain why simplifying each surd first makes it easier to tell whether a sequence is arithmetic

149

a) Write the first three terms of an arithmetic sequence with u₁ = √2 and d = 3√2 b) Write u_n

150

A geometric sequence has u₂ = 6 and u₄ = 12 with r positive. Find r and u₁ in simplest form

151

√2, √6, √18, … Show it is geometric, then find u₁₀ in simplest form

Mastery score / 15

Parent / guardian signature:

MYP 4 · Year 10 · Unit 10.2 Exponents and Indices · Practice Booklet

Homework and checking record

Hand this page in with your sheets

Name:

Your teacher will tell you which sheet and questions to do. Choose your level, set out your work in pen or pencil, then check it with the answers at the back using a green pen: circle wrong question numbers, and star ★ any you still cannot solve.

Sheet	Questions	Which ones I did	Checked in green ✓	Teacher
10.2.1 Index laws	Q1–20
10.2.1b Negative and fractional indices	Q21–38
10.2.2 Comparing exponents	Q39–57
10.2.3 Algebraic exponents	Q58–76
10.2.4 (a) Surds: what they are and simplifying	Q77–95
10.2.4 (b) Multiplying and collecting surds	Q96–114
10.2.4b Rationalising the denominator	Q115–132
10.2.5 Sequences with surds	Q133–151

Error analysis

Circle the number of any question you got wrong. Star ★ it if you still cannot solve it after correcting. Then write the questions you need to rework, and what you will do differently:

One thing I understand better now

Challenging

143Simplified: √2, 2√2, 3√2. u_n = n√2, so u₂₀ = 20√2144r = √2. u₃ = 3 × 2 = 6. u₅ = 3 × (√2)⁴ = 12145u_n = 2√3 + (n − 1)√3 = (n + 1)√3. u₈ = 9√3 = √81 × √3 = √243146She only checked the first pair. √27 ÷ √12 = 3√3 ÷ 2√3 = 1.5, not 2. The terms simplify to √3, 2√3, 3√3, so it is arithmetic with d = √31472√2 ÷ 4 = √22, and 2 ÷ 2√2 = 1√2 = √22. Both give the same ratio

Mastery

148After simplifying, the terms are multiples of the same surd, so you can see the gaps. For example 2√3, 3√3, 4√3 has a constant gap √3149a) √2, 4√2, 7√2 b) u_n = √2 + (n − 1) × 3√2 = (3n − 2)√2150u₄ = u₂ × r², so r² = 2 and r = √2. u₁ = 6√2 = 3√2151r = √3, since √6 ÷ √2 = √3. u₁₀ = √2 × (√3)⁹ = √2 × 81√3 = 81√6

Revision slides as text (easy to translate)

This is a plain-text version of the revision slides. Your browser can translate it into any language.

Slide 1: Exponents and Indices

Revision slides: everything you need for the test, one idea per slide.

Slide 2: Key vocabulary (1 of 3)

Index (power, exponent): the small number that says how many times to multiply the base, e.g. the 3 in 2³

Base: the number that is being multiplied by itself, e.g. the 2 in 2³

Negative index: an index that gives a reciprocal: 2⁻³ = 12³ = 18

Fractional index: an index that means a root: 8¹³ is the cube root of 8

Reciprocal: 1 divided by a number: the reciprocal of 8 is 18

Slide 3: Key vocabulary (2 of 3)

Exponential equation: an equation with the unknown in the index, e.g. 2^x = 32

Rational number: a number that can be written as a fraction of two integers

Irrational number: a number that cannot be written as a fraction, e.g. √2

Surd: an irrational root left in exact form, such as √2 or √12

Like surds: surds with the same root part, e.g. 3√2 and 5√2, which can be added or subtracted

Slide 4: Key vocabulary (3 of 3)

Numerator: the number or expression above the line in a fraction

Denominator: the number or expression below the line in a fraction

Rationalise the denominator: rewrite a fraction so that the denominator has no surd

Conjugate: the same two terms with the opposite sign between them: the conjugate of 3 + √2 is 3 − √2

Slide 5: Index laws (positive and zero indices)

a^m × aⁿ = a^m+n (same base: add the powers). a^m ÷ aⁿ = a^m−n.

(a^m)ⁿ = a^mn. (ab)ⁿ = aⁿbⁿ.

a⁰ = 1 because aⁿ ÷ aⁿ = a⁰ = 1.

Expand to see why: x³ × x² = (x·x·x)(x·x) = x⁵.

The laws need the same base: 2³ × 3² cannot be combined.

Slide 6: Negative and fractional indices

a⁻ⁿ = 1 ÷ aⁿ: a negative index is a reciprocal. 2⁻³ = 18.

(a/b)⁻ⁿ = (b/a)ⁿ: flip, then use the positive power.

a¹ⁿ = ⁿ√a. 9¹² = 3, 27¹³ = 3.

a^m/n = (ⁿ√a)^m. 8²³ = (³√8)² = 4.

The pattern 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 12 shows why.

Slide 7: Exponential equations

If a^x = a^y (same base, a > 0, a ≠ 1) then x = y.

Write both sides with the same base: 2^x = 32 = 2⁵, so x = 5. 9^x = 27 gives 3^2x = 3³ and x = 32.

Answers can be negative or fractional: 2^x = 18 = 2⁻³ gives x = −3.

With an algebraic exponent: 4^x+1 = 2^x+7 → 2^2x+2 = 2^x+7 → 2x + 2 = x + 7 → x = 5. Check by substituting.

The rule fails for base 1 and 0.

Slide 8: Surds

A surd is an irrational root left exact, such as √2 or √12.

Simplify using the largest square factor: √12 = √4 × √3 = 2√3. √50 = 5√2.

Add/subtract like surds only: 3√2 + 5√2 = 8√2. √2 + √3 stays as it is.

Multiply: √a × √b = √(ab), and √a × √a = a. 2√3 × 3√2 = 6√6.

Expand brackets as normal: √2(3 + √2) = 3√2 + 2. (1 + √3)² = 1 + 2√3 + 3 = 4 + 2√3.

Slide 9: Rationalising the denominator

Rationalise = remove the surd from the denominator, because a rational denominator is easier to use and compare.

Single surd: 1/√2 = 1 × √2√2 × √2 = √22. Multiplying by √2/√2 is multiplying by 1, so the value is unchanged.

Conjugate: the same terms with the opposite sign. The conjugate of 3 + √2 is 3 − √2.

13 + √2 = (3 − √2)/((3 + √2)(3 − √2)) = 3 − √29 − 2 = 3 − √27.

The product of a number and its conjugate is rational: (a + √b)(a − √b) = a² − b.

Slide 10: Sequences with surd terms

Simplify every surd term first: √2, √8, √18 = √2, 2√2, 3√2.

That is an arithmetic sequence with d = √2.

Geometric check: √2, 2, 2√2 has r = √2 (2 ÷ √2 = √2, and 2√2 ÷ 2 = √2).

Write u_n in exact form: u_n = √2 × n for the first sequence.

Slide 11: Common mistakes

✕ “x³ × x² = x⁶.”

✓ Add the powers: x⁵.

✕ “2⁻³ = −8.”

✓ A negative index gives a reciprocal: 18.

✕ “8¹³ = 83.”

✓ A power of 13 is a cube root: 2.

✕ “√2 + √3 = √5.”

✓ Unlike surds cannot be added: √2 + √3 stays as it is.

Slide 12: Common mistakes (continued)

✕ “(a + b)² = a² + b².”

✓ Expand: a² + 2ab + b². So (1 + √3)² = 4 + 2√3.