Number

Standard form

Q: What is standard form in maths?

It is a way of writing a number as A \times 10^n, where A is at least 1 and less than 10 and n is an integer (a positive or negative whole number), e.g. 4.7 \times 10^6.

Q: How do you write a small number in standard form?

Move the decimal point so there is one non-zero digit in front of it, then count how many places it moved. That number, made negative, is the power: 0.0072 = 7.2 \times 10^{-3}.

20 past-paper questions, easiest first. Try each one, then reveal the answer.

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Standard form: what you need to know

Standard form is a neat way to write very large or very small numbers, like the mass of a planet or the size of a cell. You need to convert in and out of it and calculate with it, both with and without a calculator.

Key facts

A number in standard form looks like $A \times 10^n$ , where $1 \le A < 10$ and $n$ is an integer.
Large numbers have a positive power: $52\,000 = 5.2 \times 10^4$ .
Small numbers have a negative power: $0.0031 = 3.1 \times 10^{-3}$ .
Back to an ordinary number: a positive power moves the point right, a negative power moves it left: $4.96 \times 10^{-3} = 0.00496$ .
Multiplying: multiply the $A$ parts and add the powers. Dividing: divide the $A$ parts and subtract the powers.

Multiplying and dividing

Multiply or divide the number parts.
Add (multiply) or subtract (divide) the powers of 10.
If the number part is not between 1 and 10, adjust it and change the power to match.

Adding and subtracting

Write both numbers as ordinary numbers (or with the same power of 10).
Add or subtract.
Convert the answer back to standard form.

Worked examples

Example 1 Foundation · grade 4

Write in standard form: (a) $36\,000\,000$ (b) $0.00052$

Show the solution

(a) Place the point after the first non-zero digit: $3.6$ . It moves 7 places, and the number is large.
$3.6 \times 10^7$
(b) The first non-zero digit is 5, giving $5.2$ . The point moves 4 places, and the number is small.
$5.2 \times 10^{-4}$

Example 2 Foundation · grade 5

Without a calculator, work out (a) $(6 \times 10^5) \times (4 \times 10^{-2})$ (b) $4.2 \times 10^5 + 3.1 \times 10^4$ . Give your answers in standard form.

Show the solution

(a) $6 \times 4 = 24$ and $10^5 \times 10^{-2} = 10^3$
$24 \times 10^3$ is not standard form, since 24 is more than 10.
$24 \times 10^3 = 2.4 \times 10^4$
(b) $420\,000 + 31\,000 = 451\,000$
$= 4.51 \times 10^5$

Common mistakes

Leaving answers like $24 \times 10^3$ or $0.8 \times 10^{-2}$ . The first number must be at least 1 and less than 10.
Adding the number parts and the powers separately: $4 \times 10^5 + 3 \times 10^4$ is not $7 \times 10^9$ .
Using a positive power for a small number, or moving the point the wrong way.
Ordering numbers by their powers alone when some, like $67 \times 10^3$ , are not in standard form. Convert them all to standard form first, then compare powers, then the number parts.

Exam tips

On a calculator, put brackets around each standard form number, or use the $\times 10^x$ button, so it divides by the whole number.
If the question says "give your answer in standard form", an ordinary number loses the last mark.

Questions people ask

What is standard form in maths?

It is a way of writing a number as $A \times 10^n$ , where $A$ is at least 1 and less than 10 and $n$ is an integer (a positive or negative whole number), e.g. $4.7 \times 10^6$ .

How do you write a small number in standard form?

Move the decimal point so there is one non-zero digit in front of it, then count how many places it moved. That number, made negative, is the power: $0.0072 = 7.2 \times 10^{-3}$ .

Exam questions: Standard form (20, easiest first)

Standard form2 marksGrade 4CalculatorOn both tiers

November 2024 · Paper 3 Foundation · Q18

Convert between standard form and ordinary numbers.

Show answer

(a)	$340\,200$	1 mark
(b)	$8.026 \times 10^{-1}$	1 mark

Worked solution Checked against the mark scheme

(a) Move the decimal point 5 places right: $3.402 \times 10^5 = 340\,200$

(b) Move the point 1 place right to get 8.026, so the power is $-1$ : $8.026 \times 10^{-1}$

Watch out: In (b), writing $0.8026 \times 10^0$ or using a positive power.

Standard form4 marksGrade 4CalculatorOn both tiers

June 2024 · Paper 3 Foundation · Q23

Convert between standard form and ordinary numbers, and add two numbers in standard form.

Show answer

(a(i))	$53\,000$	1 mark
(a(ii))	$0.000\,074$	1 mark
(b)	$3.42 \times 10^7$	2 marks

Worked solution Checked against the mark scheme

(a)(i) Move the decimal point 4 places right: $53\,000$

(a)(ii) Move the decimal point 5 places left: $0.000\,074$

(b) $9\,700\,000 + 24\,500\,000 = 34\,200\,000 = 3.42 \times 10^7$

Watch out: Adding the numbers in front and the powers separately, e.g. $12.15 \times 10^{13}$ .

Standard form2 marksGrade 4CalculatorOn both tiers

November 2023 · Paper 3 Foundation · Q20

Convert between ordinary numbers and standard form.

Show answer

(a)	$4.68 \times 10^5$	1 mark
(b)	0.000 503 7	1 mark

Worked solution Checked against the mark scheme

(a) Put the decimal point after the first digit: 4.68. The point moves 5 places, so $4.68 \times 10^5$ .

(b) A power of $-4$ means move the point 4 places left: $0.0005037$

Watch out: In (b), moving the point the wrong way, giving 50 370.

Standard form4 marksGrade 4CalculatorOn both tiers

November 2021 · Paper 3 Foundation · Q23

Convert between standard form and ordinary numbers, then add two numbers in standard form.

Show answer

(a)	450 000	1 mark
(b)	$7 \times 10^{-3}$	1 mark
(c)	$4.73 \times 10^3$	2 marks

Worked solution Checked against the mark scheme

(a) Move the digits 5 places: $4.5 \times 10^5 = 450\,000$

(b) $0.007 = 7 \times 10^{-3}$

Watch out: In (c), adding the numbers and powers separately to get $9.5 \times 10^5$ .

Standard form2 marksGrade 4CalculatorOn both tiers

June 2019 · Paper 2 Foundation · Q27

Convert a small decimal to standard form and a standard form number to an ordinary number.

Show answer

(a)	$5.62 \times 10^{-3}$	1 mark
(b)	$1452$	1 mark

Worked solution Checked against the mark scheme

(a) Move the decimal point 3 places right to get 5.62, so $0.00562 = 5.62 \times 10^{-3}$ .

(b) $1.452 \times 10^3$ : move the point 3 places right to get $1452$ .

Watch out: Using a positive power in (a).

Standard form2 marksGrade 4CalculatorFoundation

June 2018 · Paper 3 Foundation · Q18

Divide two numbers given in standard form and give the answer in standard form.

Show answer

Answer	$2.3 \times 10^6$	2 marks

Worked solution Checked against the mark scheme

Divide the numbers: $2.645 \div 1.15 = 2.3$ .

Subtract the powers: $10^9 \div 10^3 = 10^6$ .

Answer: $2.3 \times 10^6$ .

Watch out: Leaving the answer as 2 300 000, not in standard form.

Standard form2 marksGrade 4CalculatorHigher

Specimen · Paper 3 Higher · Q19

Convert a small decimal into standard form and a number in standard form into an ordinary number.

Show answer

(a)	$4.23 \times 10^{-4}$	1 mark
(b)	45 000	1 mark

Worked solution Checked against the mark scheme

(a) Move the decimal point 4 places right to get 4.23, so $0.000423 = 4.23 \times 10^{-4}$

(b) $4.5 \times 10^4 = 4.5 \times 10\,000 = 45\,000$

Watch out: Using a positive power in (a): 4.23 × 10⁴.

Standard form2 marksGrade 5CalculatorOn both tiers

November 2024 · Paper 2 Foundation · Q27

Solve for an unknown in a standard form equation and give it in standard form.

Show answer

Answer	$3.125 \times 10^{7}$	2 marks

Worked solution Checked against the mark scheme

$k = \frac{1.25 \times 10^{-12}}{4 \times 10^{-20}}$

$1.25 \div 4 = 0.3125$ and $10^{-12} \div 10^{-20} = 10^{8}$

$k = 0.3125 \times 10^{8} = 3.125 \times 10^{7}$

Watch out: Subtracting the powers the wrong way to get $10^{-32}$ or $10^{-8}$ .

Standard form4 marksGrade 5CalculatorFoundation

June 2023 · Paper 3 Foundation · Q28

Write a ratio of standard-form numbers as 1 : n and order numbers written in different forms.

Show answer

(a)	$1 : 50$	2 marks
(b)	$56250 \times 10^{-3}$ , $0.005625 \times 10^{5}$ , $5625$ , $5.625 \times 10^{4}$	2 marks

Worked solution Checked against the mark scheme

(a) $90\,000 : 4\,500\,000$ . Divide both by 90 000: $1 : 50$

(b) As ordinary numbers: $56\,250$ , $5625$ , $56.25$ , $562.5$

Smallest first: $56250 \times 10^{-3}$ , $0.005625 \times 10^{5}$ , $5625$ , $5.625 \times 10^{4}$

Watch out: In (b), moving the decimal point the wrong way for a negative power.

Standard form4 marksGrade 5No calculatorOn both tiers

June 2022 · Paper 1 Foundation · Q26

Convert between standard form and ordinary numbers and multiply two numbers in standard form.

Show answer

(a)	0.00163	1 mark
(b)	$4.38 \times 10^5$	1 mark
(c)	$2.4 \times 10^{-1}$	2 marks

Worked solution Checked against the mark scheme

(a) Move the decimal point 3 places left: $0.00163$

(b) $438\,000 = 4.38 \times 100\,000 = 4.38 \times 10^5$

$24 \times 10^{-2} = 2.4 \times 10^{-1}$

Watch out: Leaving (c) as 24 × 10⁻², which is not in standard form.

Standard form3 marksGrade 5CalculatorOn both tiers

November 2022 · Paper 3 Foundation · Q29

Convert standard form to an ordinary number and evaluate a calculation in standard form.

Show answer

(a)	0.000675	1 mark
(b)	$6.592 \times 10^5$	2 marks

Worked solution Checked against the mark scheme

(a) Move the decimal point 4 places left: 0.000675

(b) $2.56 \times 10^6 \times 4.12 \times 10^{-3} = 10\,547.2$

$10\,547.2 \div (1.6 \times 10^{-2}) = 659\,200 = 6.592 \times 10^5$

Watch out: Leaving the answer as 659 200 or as $65.92 \times 10^4$ .

Standard form2 marksGrade 5No calculatorOn both tiers

November 2020 · Paper 1 Foundation · Q28

Order four numbers written in standard-form style and ordinary form from smallest to largest.

Show answer

Answer	$0.000\,672$ , $67.2 \times 10^{-4}$ , $6.72 \times 10^5$ , $672 \times 10^4$	2 marks

Worked solution Checked against the mark scheme

Write each as an ordinary number:

$67.2 \times 10^{-4} = 0.00672$ , $6.72 \times 10^5 = 672\,000$ , $672 \times 10^4 = 6\,720\,000$ .

So: 0.000672, $67.2 \times 10^{-4}$ , $6.72 \times 10^5$ , $672 \times 10^4$ .

Watch out: Ordering by the powers of 10 alone and ignoring that 67.2 and 672 are not in standard form.

Standard form4 marksGrade 5CalculatorHigher

November 2020 · Paper 3 Higher · Q10

Estimate lifetime heartbeats in standard form, then divide a mass by a very large number of cells.

Show answer

(a)	$3.0 \times 10^9$ (accept $2.8 \times 10^9$ to $4.0 \times 10^9$ )	2 marks
(b)	$4.5 \times 10^{-11}$ grams	2 marks

Worked solution Checked against the mark scheme

(a) $10^5 \times 365 \times 81 = 2\,956\,500\,000 \approx 3.0 \times 10^9$

(b) $90 \div (2 \times 10^{12}) = 45 \times 10^{-12} = 4.5 \times 10^{-11}$ grams

Watch out: In (b) writing 45 × 10^-12 and not converting to standard form.

Standard form2 marksGrade 5CalculatorFoundation

November 2019 · Paper 2 Foundation · Q27

Divide two numbers in standard form and give the answer in standard form.

Show answer

Answer	$4.56 \times 10^{-2}$	2 marks

Worked solution Checked against the mark scheme

$\frac{3.42}{7.5} = 0.456$ and $10^{-7} \div 10^{-6} = 10^{-1}$

$0.456 \times 10^{-1} = 0.0456 = 4.56 \times 10^{-2}$

Watch out: Leaving the answer as $0.456 \times 10^{-1}$ , which is not in standard form.

Standard form2 marksGrade 5CalculatorHigher

November 2019 · Paper 2 Higher · Q7

Show answer

Answer	$4.56 \times 10^{-2}$	2 marks

Worked solution Checked against the mark scheme

$3.42 \div 7.5 = 0.456$

$10^{-7} \div 10^{-6} = 10^{-1}$

$0.456 \times 10^{-1} = 0.0456 = 4.56 \times 10^{-2}$

Watch out: Leaving the answer as $0.456 \times 10^{-1}$ , which is not standard form.

Standard form3 marksGrade 5CalculatorOn both tiers

November 2019 · Paper 3 Foundation · Q28

Convert to and from standard form and judge a claim comparing two numbers in standard form.

Show answer

(a)	$3.246 \times 10^7$	1 mark
(b)	0.00496	1 mark
(c)	No: $10^9$ is ten times $10^8$ , so $B$ is larger	1 mark

Worked solution Checked against the mark scheme

(a) Put the decimal point after the first digit: $3.246$ . It moves 7 places, so $3.246 \times 10^7$ .

(b) Move the point 3 places left: $0.00496$

Watch out: In (b), moving the point the wrong way to get 4960.

Standard form3 marksGrade 5CalculatorHigher

June 2019 · Paper 3 Higher · Q11

Divide numbers in standard form to find a signal's travel time, then say how a slower speed changes the answer.

Show answer

(a)	130 seconds	2 marks
(b)	The signal would take longer, so the time is an underestimate	1 mark

Worked solution Checked against the mark scheme

(a) Time $= \frac{3.9 \times 10^7}{3 \times 10^5} = 1.3 \times 10^2 = 130$ s

(b) A lower speed over the same distance means more time, so the real time is longer than 130 s.

Watch out: In (b), saying only that the answer would be 'wrong' or 'different'.

Standard form4 marksGrade 5CalculatorFoundation

November 2018 · Paper 2 Foundation · Q27

Convert between standard form and ordinary numbers, and evaluate a calculation in standard form.

Show answer

(a)	$7.547 \times 10^{-5}$	1 mark
(b)	34200	1 mark
(c)	$3.082 \times 10^{15}$	2 marks

Worked solution Checked against the mark scheme

(a) Move the decimal point 5 places right: $7.547 \times 10^{-5}$

(b) $3.42 \times 10^4 = 34\,200$

$\frac{1.541 \times 10^8}{5 \times 10^{-8}} = 0.3082 \times 10^{16} = 3.082 \times 10^{15}$

Watch out: In (a), using a positive power of 10.

Standard form3 marksGrade 5CalculatorHigher

November 2018 · Paper 2 Higher · Q7

Write a small decimal in standard form, then evaluate a calculation with numbers in standard form to 3 significant figures.

Show answer

(a)	$8.623 \times 10^{-5}$	1 mark
(b)	$7.44 \times 10^{6}$	2 marks

Worked solution Checked against the mark scheme

(a) Move the decimal point 5 places: $8.623 \times 10^{-5}$ .

(b) $\frac{3200 + 0.051}{0.00043} = \frac{3200.051}{0.00043} = 7\,441\,979.07...$

To 3 s.f. in standard form: $7.44 \times 10^6$ .

Watch out: Only dividing the second term of the numerator by the denominator (no brackets on the calculator).

Standard form4 marksGrade 6CalculatorHigher

June 2018 · Paper 3 Higher · Q9

Evaluate a formula with a square root and powers using standard form, then show that increasing both inputs actually makes the result smaller.

Show answer

(a)	$4.52 \times 10^3$	2 marks
(b)	The multiplier for $T$ is $\sqrt{1.1 \div 1.05^3} = 0.975 < 1$ , so $T$ decreases	2 marks

Worked solution Checked against the mark scheme

(a) $d^3 = (1.4 \times 10^{-4})^3 = 2.744 \times 10^{-12}$

$\frac{w}{d^3} = \frac{5.6 \times 10^{-5}}{2.744 \times 10^{-12}} = 2.0408 \times 10^{7}$

$T = \sqrt{2.0408 \times 10^7} = 4517.5... = 4.52 \times 10^3$

(b) $w$ is multiplied by 1.1 and $d^3$ by $1.05^3 = 1.1576$ . $T$ is multiplied by $\sqrt{1.1 \div 1.1576} = 0.975$ , which is less than 1, so $T$ falls (to about $4.40 \times 10^3$ ).

Watch out: Forgetting that $d$ is cubed, so its 5% increase has more effect than the 10% on $w$ .

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

Specimen · Paper 1 Higher · Q8: Estimate a planet's mass in standard form by rounding, then decide whether the estimate is too big or too small.