Indices and Standard Form

Indices are the most-taught topic in IGCSE Maths after quadratics, and the errors are remarkably consistent. Almost all of them come from two habits: multiplying the base by the power, and guessing which operation to do to the indices. Fix those two and most of the topic falls into place.

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1. What an index actually means

In aⁿ, the a is the base and the n is the index (or power, or exponent). It means a multiplied by itself n times.

• 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
• 3⁴ = 3 × 3 × 3 × 3 = 81
• 4³ = 4 × 4 × 4 = 64
• 8² = 8 × 8 = 64

This is the single most common error in the topic: 2⁵ is not 10, and 3⁴ is not 12. You do not multiply the base by the power. Every one of those examples was answered wrongly that way in real lessons. If you take one thing from this page, take this.

Worth memorising for non-calculator papers:

Squares	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Cubes	1, 8, 27, 64, 125, 216, 1000
Powers of 2	2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

Know your squares up to 20². Questions are built assuming you can recognise them instantly, and 16² = 256 (not 225 — that’s 15²).

Roots are not powers

∛27 = 3, because 3³ = 27. But 27³ = 19 683.

Confusing a cube root with cubing appeared repeatedly in lessons and is worth checking every time you read a question.

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2. The laws of indices

These only work when the bases are the same.

Law	Rule	Example
Multiplying	aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾	2⁷ × 2⁵ = 2¹²
Dividing	aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾	8⁵ ÷ 8³ = 8²
Power of a power	(aᵐ)ⁿ = a⁽ᵐⁿ⁾	(5³)⁴ = 5¹²
Zero index	a⁰ = 1	7⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	4⁻² = 1/16
Fractional index	a^(1/n) = ⁿ√a	16^(1/2) = 4
Both	a^(m/n) = (ⁿ√a)ᵐ	8^(2/3) = 4
Power of a product	(ab)ⁿ = aⁿbⁿ	(3x)² = 9x²

Multiply → add. Divide → subtract. Swapping these is the most frequent slip after the base-times-power error. A quick sanity check: 2³ × 2² is 8 × 4 = 32 = 2⁵, and 3 + 2 = 5. The rule is doing what the arithmetic does.

You never divide the indices. 10⁸ ÷ 10⁴ is 10⁴, not 10².

Zero index

Anything (except 0) to the power 0 equals 1.

7⁰ = 1, 100⁰ = 1, x⁰ = 1.

Why: 8³ ÷ 8³ = 1, because anything divided by itself is 1. But by the division law it is 8⁰. So 8⁰ must equal 1.

a⁰ is not 0, and not a. Both were common answers in lessons. Deriving it once from 8³ ÷ 8³ makes it stick.

Negative indices

a⁻ⁿ = 1/aⁿ — a negative index means reciprocal, not a negative answer.

• 4⁻² = 1/4² = 1/16
• 10⁻³ = 1/1000 = 0.001
• 2⁻¹ = 1/2

A negative power never makes the answer negative. 4⁻² is 1/16, a positive number. And 10⁻³ is not −3.

Combining: 2⁷ × 2⁻⁵ = 2⁽⁷⁻⁵⁾ = 2² = 4. Adding a negative index is the same as subtracting.

Fractional indices

The denominator is the root; the numerator is the power. a^(m/n) = (ⁿ√a)ᵐ

• 16^(1/2) = √16 = 4
• 27^(1/3) = ∛27 = 3
• 8^(2/3) = (∛8)² = 2² = 4
• 16^(3/4) = (⁴√16)³ = 2³ = 8

16^(1/2) is not 16 ÷ 2. A fractional index is a root, never a division.

Always take the root first, then apply the power — the numbers stay small. For 16^(3/4), doing ⁴√16 = 2 then 2³ = 8 is far easier than 16³ = 4096 then ⁴√4096.

With a negative fractional index, deal with the sign first:

• 81^(−1/4) = 1/81^(1/4) = 1/⁴√81 = 1/3
• 32^(−4/5) = 1/32^(4/5) = 1/(⁵√32)⁴ = 1/2⁴ = 1/16

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3. Solving equations with indices

When the bases match, the indices must be equal.

Example: solve 2ˣ = 32

• 32 = 2⁵
• so 2ˣ = 2⁵ → x = 5

Example: solve 3^(2x) = 81

• 81 = 3⁴
• 2x = 4 → x = 2

Get both sides to the same base first. That is the whole method — and it means recognising 32 as 2⁵, 81 as 3⁴, 125 as 5³, and so on.

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4. Standard form

Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.

The condition on a is what examiners check.

Number	Standard form
4 500	4.5 × 10³
87 000 000	8.7 × 10⁷
0.0032	3.2 × 10⁻³
0.000 006 1	6.1 × 10⁻⁶
35 × 10⁴	3.5 × 10⁵	not standard form as written — a must be < 10
0.6 × 10³	6 × 10²	not standard form as written — a must be ≥ 1

a must be between 1 and 10 — one non-zero digit before the decimal point. Answers like 35 × 10⁴ or 0.6 × 10³ are the right size but the wrong form, and lose the mark.

a is usually a decimal. Some students believe standard form requires a whole number; 4.5 × 10³ is perfectly correct.

Reading the sign of the power:

Big numbers → positive power. Small numbers (less than 1) → negative power.

The power tells you how many places the decimal point moves: right for positive, left for negative.

A negative power does not mean a negative number. 3.2 × 10⁻³ = 0.0032, which is positive. It is a small number, not a negative one.

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5. Calculating with standard form

Multiplying and dividing — straightforward

Deal with the numbers and the powers separately.

(3 × 10⁵) × (4 × 10³)

• 3 × 4 = 12
• 10⁵ × 10³ = 10⁸
• = 12 × 10⁸ = 1.2 × 10⁹ ← convert back to standard form

(8 × 10⁷) ÷ (2 × 10⁴)

• 8 ÷ 2 = 4
• 10⁷ ÷ 10⁴ = 10³
• = 4 × 10³

Check the final answer is still in standard form. 12 × 10⁸ must become 1.2 × 10⁹. This last step is skipped constantly.

Adding and subtracting — the one that catches people

You cannot add or subtract in standard form unless the powers of 10 are the same.

(4 × 10⁵) + (3 × 10⁴)

Make the powers match:

• 3 × 10⁴ = 0.3 × 10⁵
• 4 × 10⁵ + 0.3 × 10⁵ = 4.3 × 10⁵
• = 4.3 × 10⁵

Or convert both to ordinary numbers, add, and convert back: 400 000 + 30 000 = 430 000 = 4.3 × 10⁵.

Adding the numbers and keeping the larger power is wrong. (4 × 10⁵) + (3 × 10⁴) is not 7 × 10⁵. Match the powers first, every time.

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6. Prime factorisation, HCF and LCM in index form

Writing a number as a product of its prime factors underpins HCF, LCM, and questions about squares and cubes.

360 = 2³ × 3² × 5

HCF — take the lowest power of each common prime. LCM — take the highest power of every prime that appears.

Example: 360 = 2³ × 3² × 5 and 84 = 2² × 3 × 7

• HCF = 2² × 3 = 12
• LCM = 2³ × 3² × 5 × 7 = 2520

Two results worth knowing:

A number is a perfect square when every prime factor has an even power. A number is a perfect cube when every prime factor has a power that is a multiple of 3.

These turn “what is the smallest number you must multiply 360 by to make it a square?” into a quick check: 360 = 2³ × 3² × 5 has odd powers on 2 and 5, so multiply by 2 × 5 = 10.

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7. Calculators — and a warning

Tutors disagreed on this in real lessons, and it matters:

Check which of your papers allow a calculator. The 0580 assessment structure has changed in recent years and now includes both calculator and non-calculator papers. Confirm against your own exam timetable and syllabus version rather than relying on a paper number you heard somewhere — advice given in lessons was contradictory on exactly this point.

When a calculator is allowed:

• Use brackets round negative numbers and round whole fractional indices
• Use the S⇔D key to switch between fraction and decimal form
• Still write down your working — a bare answer earns fewer marks, and method marks are only available if the method is visible

When it isn’t:

• Know your squares, cubes and powers of 2
• Convert decimals to fractions — 0.25^(−1) is much easier as (1/4)^(−1) = 4
• Take roots before powers to keep numbers small

Don’t reach for the calculator on ∛27. Recognising it as 3 is faster and less error-prone, and on a non-calculator paper it is the only option.

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8. Mistakes that cost marks

Multiplying the base by the power. 2⁵ = 32, not 10.

Adding indices when dividing, or subtracting when multiplying.

Dividing the indices. You never do this.

Saying a⁰ = 0 or a⁰ = a. It is 1.

Treating a negative index as a negative answer. 4⁻² = 1/16.

Reading 10⁻³ as −3.

Treating a fractional index as a division. 16^(1/2) = 4, not 8.

Confusing a cube root with a cube. ∛27 = 3.

Leaving an answer as 35 × 10⁴, which is not standard form.

Adding numbers in standard form without matching the powers of 10.

Forgetting to convert back to standard form at the end.

Applying a power to only part of a product. (3x)² = 9x², not 3x².

Using index laws on different bases. 2³ × 3² cannot be combined.

Rounding too early in multi-step standard form calculations.

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Frequently asked questions

What does 2⁵ mean? 2 multiplied by itself 5 times = 32. It does not mean 2 × 5.

What is the rule for multiplying indices? Same base: add the powers. aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾.

What is anything to the power of 0? 1 — for any non-zero base.

What does a negative index mean? The reciprocal: a⁻ⁿ = 1/aⁿ. The answer stays positive.

What does a fractional index mean? A root. The denominator gives the root, the numerator the power: a^(m/n) = (ⁿ√a)ᵐ.

What is 8^(2/3)? Cube root of 8 is 2, then squared = 4.

What is standard form? A number written as a × 10ⁿ with 1 ≤ a < 10 and n an integer.

Is 35 × 10⁴ in standard form? No — a must be less than 10. Correctly written it is 3.5 × 10⁵.

How do you add numbers in standard form? Make the powers of 10 the same first, then add the numbers. Or convert to ordinary numbers, add, and convert back.

How do you find the HCF and LCM using indices? Write both as products of primes. HCF: lowest power of each common prime. LCM: highest power of every prime present.

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Quick revision checklist

• I know aⁿ means repeated multiplication, not a × n
• I know my squares to 20², cubes, and powers of 2
• I can tell a cube root from a cube
• I add indices when multiplying and subtract when dividing
• I multiply indices for a power of a power
• I know a⁰ = 1 and can explain why
• I know a⁻ⁿ = 1/aⁿ and that the answer stays positive
• I can evaluate fractional indices, root first
• I can handle negative and fractional indices together
• I can solve equations by making the bases the same
• I know standard form needs 1 ≤ a < 10
• I can convert both ways, including small numbers with negative powers
• I can multiply and divide in standard form and convert back
• I match the powers before adding or subtracting
• I can write a number as a product of primes in index form
• I can find HCF and LCM from prime factorisation
• I know the even-power test for squares and multiple-of-3 test for cubes
• I have checked which of my papers allow a calculator

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These notes cover indices and standard form in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus, formula list and calculator rules for your own exam series.