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Edexcel IGCSE Maths (4MA1) · Unit 1: Number

Edexcel IGCSE 4MA1 Indices and Standard Form: exam questions and answers, notes and worked examples

Q: Are the laws of indices on the 4MA1 formula sheet?

No. The laws of indices and standard form are not printed on the 4MA1 Higher formulae sheet, so learn them by heart.

Q: What does a fractional index mean?

The denominator is a root and the numerator is a power. For 27 to the power 2/3, take the cube root of 27 (3) and then square it (9). Taking the root first keeps the numbers small.

Q: Can I use a calculator for indices and standard form in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

Q: How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Updated October 2026 · By Pete Bromfield

The laws of indices tell you what to do with powers of the same base: add the indices when you multiply, subtract them when you divide, and multiply them when you raise a power to a power. A negative index means a reciprocal and a fractional index means a root, so $16^{\frac{3}{4}} = 8$ . Standard form writes a number as $A \times 10^{n}$ with $1 \leq A < 10$ , which is how you work with very large and very small numbers.

Unit 1: Number
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Indices and Standard Form notes

The laws of indices

For the same base: $a^{m} \times a^{n} = a^{m + n}$ , $a^{m} \div a^{n} = a^{m - n}$ , $(a^{m})^{n} = a^{m n}$ and $a^{0} = 1$ . Deal with the numbers in front separately from the powers: $3 x^{4} \times 5 x^{2} = 15 x^{6}$ and $(2 x^{3})^{4} = 16 x^{12}$ , because the 2 is raised to the power 4 as well.

Negative and fractional indices

A negative index is a reciprocal: $a^{- n} = \frac{1}{a^{n}}$ , so $5^{- 2} = \frac{1}{25}$ . A fractional index is a root: $a^{\frac{1}{n}}$ is the $n$ th root of $a$ , and $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ , so $27^{\frac{2}{3}} = 9$ . With a negative fraction, flip first and then take the root: ${(\frac{4}{9})}^{- \frac{1}{2}} = \frac{3}{2}$ .

Solving equations with powers

Write both sides as powers of the same number and compare the indices. $2^{x} = 32$ gives $x = 5$ , because $32 = 2^{5}$ . For $9^{x} = 27$ , write $9 = 3^{2}$ and $27 = 3^{3}$ : $3^{2 x} = 3^{3}$ , so $2 x = 3$ and $x = \frac{3}{2}$ .

Writing numbers in standard form

The first number must be at least 1 and less than 10. Count how many places the decimal point moves: 340 000 $= 3.4 \times 10^{5}$ , and $0.0062 = 6.2 \times 10^{- 3}$ . Large numbers have a positive power of 10 and numbers between 0 and 1 have a negative one.

Calculating in standard form

To multiply, multiply the first numbers and add the powers, then tidy up: $(3 \times 10^{4}) \times (5 \times 10^{6}) = 15 \times 10^{10} = 1.5 \times 10^{11}$ . To divide, divide and subtract the powers: $(8 \times 10^{3}) \div (2 \times 10^{- 2}) = 4 \times 10^{5}$ . To add or subtract, use the same power of 10 first: $4.5 \times 10^{5} + 3 \times 10^{4} = 4.8 \times 10^{5}$ .

Key formulas

Indices Learn it: $a^{m} \times a^{n} = a^{m + n}, a^{m} \div a^{n} = a^{m - n}, (a^{m})^{n} = a^{m n}, a^{0} = 1, a^{- n} = \frac{1}{a^{n}}, a^{\frac{m}{n}} = (\sqrt[n]{a})^{m}$
Standard form Learn it: $A \times 10^{n}, 1 \leq A < 10, n \in Z$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Indices and Standard Form worked examples

Worked example 1: Fractional indices

Evaluate $32^{\frac{3}{5}}$ .

Solution

The denominator 5 means fifth root: $\sqrt[5]{32} = 2$ .
Then cube: $2^{3} = 8$ .

3 steps · 1 to fill in

Tip: Root first, then power, keeps the numbers small.

Worked example 2: Converting to standard form

Write (a) 0.000 070 3 in standard form (b) $4.6 \times 10^{6}$ as an ordinary number.

Solution

(a) Move the decimal point 5 places right to get 7.03, so $7.03 \times 10^{- 5}$ .
(b) Move the decimal point 6 places right: 4 600 000.

3 steps

Tip: Small numbers have negative powers of 10.

Worked example 3: Calculating in standard form

Work out $(7 \times 10^{5}) \times (6 \times 10^{- 2})$ , giving your answer in standard form.

Solution

Multiply the numbers: $7 \times 6 = 42$ .
Add the powers: $10^{5 + (- 2)} = 10^{3}$ .
$42 \times 10^{3} = 4.2 \times 10^{4}$ .

4 steps

Tip: The first number must be between 1 and 10.

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Indices and Standard Form exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

Simplify $(3 x^{4} y^{2})^{3}$ .

Show the answer and mark scheme

B2 27x¹²y⁶ (B1 for two of 27, x¹², y⁶ in a product)

Answer: $27 x^{12} y^{6}$

Question 2 · easy · 2 marks

(a) Write 0.000 307 in standard form. (b) Write $5.6 \times 10^{5}$ as an ordinary number.

Show the answer and mark scheme

B1 3.07 × 10⁻⁴
B1 560 000

Answer: (a) $3.07 \times 10^{- 4}$ (b) 560 000

Question 3 · medium · 2 marks

Simplify fully $\frac{6 x^{5} y^{3} \times 4 x^{2} y}{8 x^{3} y^{2}}$ .

Show the answer and mark scheme

M1 numerator 24x⁷y⁴, or two of 3, x⁴, y² in the answer
A1 3x⁴y² cao

Answer: $3 x^{4} y^{2}$

Question 4 · medium · 3 marks

Work out the value of (a) $64^{\frac{2}{3}}$ (b) ${(\frac{27}{8})}^{- \frac{1}{3}}$

Show the answer and mark scheme

B1 16
M1 8/27 or 3/2 seen
A1 2/3 oe

Answer: (a) 16 (b) $\frac{2}{3}$

Question 5 · medium · 2 marks

Write these numbers in order of size. Start with the smallest. $4.2 \times 10^{- 3}$ , 0.0039, $41 \times 10^{- 4}$ , $4.02 \times 10^{- 3}$

Full worked solution Free, no sign-in

B2 0.0039, 4.02 × 10⁻³, 41 × 10⁻⁴, 4.2 × 10⁻³ (B1 for all four written in the same form, or three in the correct order)

Answer: 0.0039, $4.02 \times 10^{- 3}$ , $41 \times 10^{- 4}$ , $4.2 \times 10^{- 3}$

Every question here has a full worked solution. See them all →

Question 6 · medium · 3 marks

$p = 6 \times 10^{7}$ and $q = 4 \times 10^{- 3}$ . Give each answer in standard form. (a) Work out $p q$ . (b) Work out $p \div q$ .

Show the answer and mark scheme

M1 24 × 10⁴ oe
A1 2.4 × 10⁵
B1 1.5 × 10¹⁰

Answer: (a) $2.4 \times 10^{5}$ (b) $1.5 \times 10^{10}$

Question 7 · hard · 3 marks

Find the value of $x$ for which $8^{x + 1} = 4^{2 x - 1}$ .

Show the answer and mark scheme

M1 both sides as powers of 2: 2^(3x + 3) = 2^(4x − 2)
M1 3x + 3 = 4x − 2
A1 x = 5

Answer: $x = 5$

Question 8 · hard · 3 marks

A grain of sand has a mass of $6.4 \times 10^{- 4}$ grams. A bag holds 2.5 kg of this sand. Work out the number of grains in the bag. Give your answer in standard form correct to 3 significant figures.

Show the answer and mark scheme

M1 2.5 kg = 2500 g
M1 2500 ÷ (6.4 × 10⁻⁴)
A1 3.91 × 10⁶ (3 906 250)

Answer: $3.91 \times 10^{6}$

Common indices and standard form mistakes

Multiplying the bases. $2^{3} \times 2^{4} = 2^{7}$ , not $4^{7}$ . The base stays the same; only the indices change.
Reading a negative index as a negative number. $4^{- 2} = \frac{1}{16}$ , not $- 16$ or $- 8$ . A negative index gives a reciprocal.
A first number of 10 or more. $15 \times 10^{10}$ is not in standard form. Divide 15 by 10 and add 1 to the power: $1.5 \times 10^{11}$ .
Adding the powers when you add. Only multiplying adds the powers. To add or subtract numbers in standard form, write them with the same power of 10 first, or as ordinary numbers.

Where students lose marks across 4MA1, from the examiner reports →

Indices and Standard Form FAQ

Are the laws of indices on the 4MA1 formula sheet?

No. The laws of indices and standard form are not printed on the 4MA1 Higher formulae sheet, so learn them by heart.

What does a fractional index mean?

The denominator is a root and the numerator is a power. For 27 to the power 2/3, take the cube root of 27 (3) and then square it (9). Taking the root first keeps the numbers small.

Can I use a calculator for indices and standard form in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Indices and Standard Form revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 indices questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Indices and Standard Form: exam questions and answers, notes and worked examples

Try it first

Indices and Standard Form notes

The laws of indices

Negative and fractional indices

Solving equations with powers

Writing numbers in standard form

Calculating in standard form

Key formulas

Indices and Standard Form worked examples

Worked example 1: Fractional indices

Solution

Worked example 2: Converting to standard form

Solution

Worked example 3: Calculating in standard form

Solution

Indices and Standard Form exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · medium · 2 marks

Question 4 · medium · 3 marks

Question 5 · medium · 2 marks

Full worked solution Free, no sign-in

Question 6 · medium · 3 marks

Question 7 · hard · 3 marks

Question 8 · hard · 3 marks

Common indices and standard form mistakes

Indices and Standard Form FAQ

Next steps

Practise indices and standard form with your progress saved

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