Overview
What you’ll learn
- Apply the index laws to multiply, divide and raise powers, including zero and negative indices.
- Evaluate fractional indices exactly, e.g. 27^(2/3) and 16^(−1/2) (Extended).
- Convert numbers between ordinary form and standard form A × 10ⁿ with 1 ⩽ A < 10.
- Multiply and divide numbers in standard form, adjusting the exponent so 1 ⩽ A < 10.
- Simplify surds and expressions like √50 + √18, and rationalise denominators (Extended).
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 1.7Understand and use indices with positive, zero and negative integer powers, and use the rules of indices.
- 1.8Use standard form A × 10ⁿ where n is a positive or negative integer and 1 ⩽ A < 10, and convert numbers into and out of standard form.
- 1.8Calculate with values in standard form (Core candidates are expected to calculate with standard form only on Paper 3).
Extended — Extended candidates only
- 1.7Understand and use fractional indices, e.g. 16^(1/2) and 8^(1/3), alongside the index laws. (Extended)
- 1.18Understand and use surds, including simplifying expressions such as √20 = 2√5 and √200 − √32 = 6√2. (Extended)
- 1.18Rationalise the denominator of a fraction involving a surd. (Extended)
In 30 seconds
- 1Multiply powers: add the indices (2⁻³ × 2⁴ = 2); divide: subtract (3⁴ ÷ 3⁶ = 3⁻²); raise a power: multiply ((2³)² = 2⁶).
- 2Anything (non-zero) to the power zero is 1; a negative index flips into a fraction: 5⁻² = 1/25.
- 3Extended — fractional indices are roots: the denominator is the root, the numerator the power: 27^(2/3) = (∛27)² = 9.
- 4Standard form is A × 10ⁿ with 1 ⩽ A < 10: 3 400 = 3.4 × 10³ and 0.00042 = 4.2 × 10⁻⁴.
- 5Multiplying in standard form: multiply the A-values, add the exponents, then re-balance so 1 ⩽ A < 10.
- 6Surds keep answers exact (Extended): √20 = 2√5; add like surds (√50 + √18 = 8√2); rationalise by multiplying top and bottom by the surd (or its conjugate).
- 7On non-calculator papers, exact forms — fractions, powers, surds — are the expected answers; never round to decimals unless told to.
Notes · 1 of 3
The index laws
Multiply, divide and raise powers — with zero, negative and fractional indices.
When powers of the same base are multiplied, the indices add: 2⁻³ × 2⁴ = 2¹ = 2. When divided, they subtract: 3⁴ ÷ 3⁶ = 3⁻² = 1/9. When a power is raised, the indices multiply: (2³)² = 2⁶ = 64. These three laws carry everything else in this topic — including letters later in the algebra topics.
The zero index gives 1 every time: 7⁰ = 1 (and later, x⁰ = 1 for any non-zero x). The negative index is a reciprocal: 5⁻² = 1/5² = 1/25, and 2⁻³ = 1/8. It never makes the answer negative — that is the single most common slip.
Extended. A fractional index is a root: the denominator says which root, the numerator the power. So 16^(1/2) = 4, 8^(1/3) = 2, 27^(2/3) = (∛27)² = 9, and negative-fractional combinations work the same: 16^(−1/2) = 1/4 and 8^(−2/3) = 1/4.
Notes · 2 of 3
Standard form
A × 10ⁿ with 1 ⩽ A < 10 — writing, converting and calculating.
Standard form writes a number as A × 10ⁿ where 1 ⩽ A < 10 and n is an integer. Big numbers get positive exponents: 3 400 = 3.4 × 10³. Small numbers get negative ones: 0.00042 = 4.2 × 10⁻⁴. The A-value must sit between 1 and 10 — 34 × 10² is *not* standard form until it becomes 3.4 × 10³.
| Ordinary form | Standard form |
|---|---|
| 7 900 000 000 | 7.9 × 10⁹ |
| 605 000 | 6.05 × 10⁵ |
| 0.0002 | 2 × 10⁻⁴ |
| 0.00000703 | 7.03 × 10⁻⁶ |
Calculating in standard form: multiply the A-values and add the exponents; divide and subtract. Then re-balance. For (3.2 × 10⁵) × (4 × 10³): 3.2 × 4 = 12.8 and 10⁵ × 10³ = 10⁸, giving 12.8 × 10⁸ = 1.28 × 10⁹. For (6 × 10⁸) ÷ (3 × 10³): 6 ÷ 3 = 2 and 10⁸⁻³ = 10⁵, so 2 × 10⁵ — already in standard form. Core candidates only need this arithmetic on the calculator paper, but the conversion skill is universal.
Notes · 3 of 3
Surds and rationalising denominators
Keeping roots exact: simplifying, collecting like surds, cleaning the denominator. (Extended)
A surd is a root that cannot be simplified to a whole number or fraction — √2, √5, ∛9. Surds keep answers exact, which the non-calculator Extended papers prize. Simplify using the largest square factor: √20 = √(4 × 5) = 2√5, and √48 = √(16 × 3) = 4√3.
Surds behave like algebraic terms once simplified: √50 + √18 = 5√2 + 3√2 = 8√2, while √200 − √32 = 10√2 − 4√2 = 6√2. You can only add or subtract like surds — √2 + √3 stays as it is, and √2 × √3 = √6 follows the multiplication law.
Rationalising a denominator removes the surd from underneath a fraction. For a single surd, multiply top and bottom by it: 10/√5 = 10√5/5 = 2√5; similarly 6/√3 = 2√3. For a two-term denominator like 3 − √2, multiply by the conjugate 3 + √2: the difference of two squares turns the denominator into 9 − 2 = 7, so 1/(3 − √2) = (3 + √2)/7.
Revise
Key terms
The words examiners expect you to use precisely.
- Index (power)
- The small raised number showing how many times a base multiplies itself.In 2⁵ the index is 5.
- Index laws
- The rules for multiplying, dividing and raising powers of the same base.Add, subtract and multiply the indices respectively.
- Negative index
- A power that means reciprocal: a⁻ⁿ = 1/aⁿ.5⁻² = 1/25.
- Fractional index (Extended)
- A power whose denominator is a root: a^(1/n) = ⁿ√a.27^(2/3) = 9.
- Standard form
- A number written as A × 10ⁿ with 1 ⩽ A < 10 and integer n.0.00042 = 4.2 × 10⁻⁴.
- Surd (Extended)
- An irrational root left in exact form, such as √5.√20 simplifies to 2√5, still a surd.
Show 4 more key termsShow fewer key terms
- Like surds (Extended)
- Surds with the same radicand, which can be collected.5√2 + 3√2 = 8√2.
- Rationalise (Extended)
- Rewriting a fraction so its denominator contains no surd.10/√5 = 2√5.
- Conjugate (Extended)
- The matching two-term expression with the surd sign flipped.(3 − √2) and (3 + √2) multiply to 7.
- Exact value
- An answer left as a fraction, power or surd rather than a decimal.Expected on non-calculator papers.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Multiplying the indices when multiplying powers of different bases.
The laws apply to a shared base only.
Instead: 2³ × 3² has no shortcut — evaluate each power, then multiply.
Mistake: Reading 5⁻² as −25.
The negative belongs to the index, not the answer.
Instead: 5⁻² = 1/5² = 1/25, always positive here.
Show 11 more mistakesShow fewer mistakes
Mistake: Adding indices when raising a power to a power.
(2³)² is 2⁶, not 2⁵.
Instead: Raising multiplies the indices: 3 × 2 = 6.
Mistake: Treating 7⁰ as 0 or 7.
The zero-index law fixes it at exactly 1.
Instead: a⁰ = 1 for any non-zero base.
Mistake: Writing standard form with A ⩾ 10.
12.8 × 10⁸ breaks the 1 ⩽ A < 10 rule.
Instead: Rebalance to 1.28 × 10⁹ by shifting one exponent.
Mistake: Adding the A-values when dividing in standard form.
Division divides the A-values and subtracts the exponents.
Instead: (6 × 10⁸) ÷ (3 × 10³) = 2 × 10⁵.
Mistake: Adding unlike surds such as √2 + √3.
Only like surds collect.
Instead: Simplify each surd first; √2 + √3 cannot combine.
Mistake: Simplifying √20 as 4√5 (halving instead of factoring).
√(ab) = √a × √b needs a square factor: 20 = 4 × 5.
Instead: √20 = 2√5, pulling out the square root of the square factor.
Mistake: Multiplying only the numerator when rationalising.
Multiplying the top by √5 alone changes the value.
Instead: Multiply top and bottom by the same surd (or conjugate).
Mistake: Using the wrong conjugate, e.g. (3 + √2) becomes (3 − √2) twice.
The conjugate flips the sign of the surd term only.
Instead: 1/(3 − √2) × (3 + √2)/(3 + √2) = (3 + √2)/7.
Mistake: Leaving 30.8 × 10⁻⁷ as a standard-form answer.
The front number must be at least 1 and under 10.
Instead: Rewrite 30.8 as 3.08 × 10¹ and combine the powers: 3.08 × 10⁻⁶.
Mistake: Evaluating 4^(3/2) as 4 × 3 ÷ 2.
The denominator of a fractional index is a root, not a division of the whole calculation.
Instead: 4^(3/2) = (√4)³ = 2³ = 8.
Mistake: Squaring only part of 2√5.
Both the 2 and the surd must be squared.
Instead: (2√5)² = 2² × (√5)² = 4 × 5 = 20.
Check yourself
Quick quiz
24 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Multiply powers of the same base
Work out 2⁻³ × 2⁴.
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 4 Practice Paper
30 marks · 40 min · 4 pages
10 multiple-choice questions and structured questions with answer lines. Topic 4: index laws with zero, negative and fractional indices, standard form conversions and arithmetic, simplifying surds, and rationalising denominators.
Mark scheme
PROTopic 4 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
Why must my standard form answer have A between 1 and 10?
That is the convention that makes standard form unique: every number has exactly one standard-form representation. 12.8 × 10⁸ is the right value but not standard form — shift one place to get 1.28 × 10⁹.
What order should I work out fractional indices like 27^(2/3)?
Root first, power second. The cube root of 27 is 3, and 3² = 9. Doing it that way keeps every number small; multiplying first (27²) makes the arithmetic — and the errors — much bigger.
When can I add surds?
Only after simplifying, and only when they are the same kind of surd. √50 + √18 works because both collapse to multiples of √2. √2 + √3 stays as it is — there is no law to combine different surds by addition.
Why do I have to rationalise denominators?
Convention, and clarity: an exact answer with a rational denominator is easier to compare and to use in further working. Multiply top and bottom by the surd — or, for two-term denominators, by its conjugate.
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