Standard form | OCR GCSE Maths Notes

Visual model

Standard form writes huge or tiny numbers neatly

10^{-3}

10^{0}

10^{3}

1 \le A < 10

Gold-standard guide

20 mins

What you will learn

Write and calculate with very large or very small numbers.

Use a clear step-by-step method for standard form.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier number skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Multiply the numbers and add the powers

Step 1

Multiply the ordinary numbers together

3.2 × 2.5 = 8.0

Step 2

Multiply the powers of 10 by adding the indices

10⁴ × 10³ = 10^(4+3) = 10⁷

Step 3

Combine and check the format

8.0 × 10⁷

Watch out

Students write 8.0 × 10⁷ as 80 × 10⁶ — this is NOT standard form because 80 is not between 1 and 10

Standard form

A x 10ⁿ where 1 ≤ A < 10.

Multiplying

$(a \times 10^m)(b \times 10^n) = ab \times 10^(m+n).$

Worked example

Without a calculator, work out (3.2 × 10⁴) × (2.5 × 10³), giving your answer in standard form

0. Work these out separately from the powers of 10 because it keeps the arithmetic manageable.

Multiply the powers of 10 by adding the indices: 10⁴ × 10³ = 10^(4+3) = 10⁷.We add because we are multiplying the same base.

0 × 10⁷. The first part is 8, which is between 1 and 10, so this is correct standard form.

Final answer

8.0 × 10⁷

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Without a calculator, work out (3.2 × 10⁴) × (2.5 × 10³), giving your answer in standard form

3 marks4 minsstandard-form-worked

Show solution

Worked solution

1.0. Work these out separately from the powers of 10 because it keeps the arithmetic manageable.
2.Multiply the powers of 10 by adding the indices: 10⁴ × 10³ = 10^(4+3) = 10⁷.We add because we are multiplying the same base.
3.0 × 10⁷. The first part is 8, which is between 1 and 10, so this is correct standard form.

Final answer

8.0 × 10⁷

Mark points

M1: multiply the ordinary numbers together
M1: multiply the powers of 10 by adding the indices
M1: combine and check the format
A1: 8.0 × 10⁷

Watch out

0 × 10⁷ as 80 × 10⁶ — this is NOT standard form because 80 is not between 1 and 10.Always check: the coefficient must be at least 1 and less than 10.If it is not, adjust: if the front number is ≥ 10, divide it by 10 and add 1 to the power.

Diagnosticrecall

Write 0.000 047 in standard form

1 mark2 minsstandard-form-q1

Show solution

Worked solution

1.Spot the skill: Multiply the numbers and add the powers.
2.Use the multiply the ordinary numbers together stage first, then multiply the powers of 10 by adding the indices.
3.Keep the final answer visible: 4.7 × 10⁻⁵.

Final answer

4.7 × 10⁻⁵

Mark points

M1: use the correct multiply the numbers and add the powers.because 10ⁿ × 10ᵐ = 10^(n+m), you can deal with the powers of 10 separately from the ordinary numbers.check the front number is between 1 and 10 at the end.
A1: 4.7 × 10⁻⁵

Watch out

Easyprocedure

Write 6.03 × 10⁵ as an ordinary number

2 marks3 minsstandard-form-q2

Show solution

Worked solution

1.Spot the skill: Multiply the numbers and add the powers.
2.Use the multiply the powers of 10 by adding the indices stage first, then combine and check the format.
3.Keep the final answer visible: 603 000.

Final answer

603 000

Mark points

M1: use the correct multiply the numbers and add the powers.because 10ⁿ × 10ᵐ = 10^(n+m), you can deal with the powers of 10 separately from the ordinary numbers.check the front number is between 1 and 10 at the end.
A1: 603 000

Watch out

Mediumreasoning

Work out (5 × 10³) + (3 × 10²), giving your answer in standard form

3 marks4 minsstandard-form-q3

Show solution

Worked solution

1.Spot the skill: Multiply the numbers and add the powers.
2.Use the combine and check the format stage first, then multiply the ordinary numbers together.
3.Keep the final answer visible: 5.3 × 10³.

Final answer

5.3 × 10³

Mark points

M1: use the correct multiply the numbers and add the powers.because 10ⁿ × 10ᵐ = 10^(n+m), you can deal with the powers of 10 separately from the ordinary numbers.check the front number is between 1 and 10 at the end.
A1: 5.3 × 10³

Watch out

Hardproblem solving

Work out (9.6 × 10⁸) ÷ (3 × 10³), giving your answer in standard form

3 marks5 minsstandard-form-q4

Show solution

Worked solution

1.Spot the skill: Multiply the numbers and add the powers.
2.Use the multiply the ordinary numbers together stage first, then multiply the powers of 10 by adding the indices.
3.Keep the final answer visible: 3.2 × 10⁵.

Final answer

3.2 × 10⁵

Mark points

M1: use the correct multiply the numbers and add the powers.because 10ⁿ × 10ᵐ = 10^(n+m), you can deal with the powers of 10 separately from the ordinary numbers.check the front number is between 1 and 10 at the end.
A1: 3.2 × 10⁵

Watch out

Exam-stylemulti-step

The distance from Earth to the Sun is 1.5 × 10⁸ km. Light travels at 3 × 10⁵ km/s. How many seconds does light take to reach Earth from the Sun? Give your answer in standard form.

4 marks6 minsstandard-form-q5

Show solution

Worked solution

1.Spot the skill: Multiply the numbers and add the powers.
2.Use the multiply the powers of 10 by adding the indices stage first, then combine and check the format.
3.Keep the final answer visible: 5 × 10² seconds.

Final answer

5 × 10² seconds

Mark points

M1: use the correct multiply the numbers and add the powers.because 10ⁿ × 10ᵐ = 10^(n+m), you can deal with the powers of 10 separately from the ordinary numbers.check the front number is between 1 and 10 at the end.
A1: 5 × 10² seconds

Watch out

Grade 9 stretchproblem solving

Work out ( $6 \times 10^{7}$ )( $4 \times 10^{-3}$ ) / ( $8 \times 10^{2}$ ). Give your answer in standard form.

4 marks7 minsstandard-form-g9

Show solution

Worked solution

1.Calculate the number parts.
2.Use the index laws on the powers of 10.
3.Check the first number is at least 1 and less than 10.

Final answer

$3 \times 10^{2}$

Mark points

M1: use 6 × 4 / 8
M1: use 10^(7 - 3 - 2)
A1: $3 \times 10^{2}$

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Hard exam-stylemulti-step problem

A spacecraft travels $7.2 \times 10^{8}$ metres in $4.5 \times 10^{3}$ seconds. Work out its average speed in standard form.

3 marks6 minsstandard-form-paper

Show solution

Worked solution

1.Divide the number parts.
2.Subtract the powers of 10.
3.Check the first number is at least 1 and less than 10.

Final answer

$1.6 \times 10^{5}$ m/s

Mark points

M1: use 7.2 / 4.5
M1: use 10^(8 - 3)
A1: obtain $1.6 \times 10^{5}$ m/s

Watch out

Read the full question before calculating. Keep each stage of your working visible.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Standard form - 2 marksWrite 0.000 047 in standard formMark answer

Answer

4.7 × 10⁻⁵

2Calculations and order of operations - 2 marksWork out (6 + 2) × 3 − 5Mark answer

Answer

3Integers, decimals and place value - 2 marksOrder these from smallest to largest: 0.3, 0.03, 0.303, 0.033Mark answer

Answer

0.03, 0.033, 0.3, 0.303

4Fractions - 3 marksA recipe needs ¾ cup of sugar. How much sugar is needed for 2½ batches?Mark answer

Answer

1⅞ cups

Mastery check

I can explain the method for standard form.
I can show clear working without skipping key steps.
0 × 10⁷ as 80 × 10⁶ — this is NOT standard form because 80 is not between 1 and 10.Always check: the coefficient must be at least 1 and less than 10.If it is not, adjust: if the front number is ≥ 10, divide it by 10 and add 1 to the power.

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