Is 1 10^3 in standard form?

Yes. A = 1 is allowed because the condition is 1 A < 10.

1MA1 · N9NumberFoundation & HigherGrades 4–7

Standard Form

Q: Why must A be less than 10?

It makes every number have exactly one standard form, so the power of 10 always shows the order of magnitude.

The distance to the Sun is about $150000000$ km; a red blood cell is about $0.000 008$ m across. Writing out all those zeros is slow and error-prone. $Standard form$ writes any number as a value between 1 and 10 multiplied by a power of 10 — so those become $1.5 \times 1 0^{8}$ km and $8 \times 1 0^{- 6}$ m.

45 min Powers, Roots & Standard Form

The big picture

Standard form separates a number into its digits ( $A$ ) and its size ( $1 0^{n}$ ). That makes very different numbers easy to compare — the power of 10 tells you the order of magnitude at a glance — and it turns multiplication and division into simple work with index laws.

Edexcel sets standard form on both calculator and non-calculator papers, frequently in science or population contexts, and expects answers written in correct standard form, not just any power of 10.

What you'll learn

Convert between ordinary numbers and standard form
Order numbers written in standard form
Multiply and divide in standard form using index laws
Add and subtract numbers in standard form

Writing numbers in standard form

Standard form is $A \times 1 0^{n}$ where $1 \leq A < 10$ and $n$ is an integer.

For large numbers, $n$ is positive: count how many places the digits move to make $A$ . $47000 = 4.7 \times 1 0^{4}$ .

For numbers less than 1, $n$ is negative: $0.000 36 = 3.6 \times 1 0^{- 4}$ .

1(a)

A = 6.02

; digits move 8 places.

2(b)

A = 5.07

; digits move 5 places the other way.

Answer(a)

6.02 \times 1 0^{8}

(b)

5.07 \times 1 0^{- 5}

Your turnSame method, new numbers — try it before moving on.

Write $0.000 309$ in standard form.

Your answer

Type it like 4.2 x 10^-3.

Tip — Check $A$ first: it must be at least 1 and less than 10. $45 \times 1 0^{3}$ and $0.45 \times 1 0^{5}$ both equal $45000$ but neither is in standard form.

Ordering

Compare the powers of 10 first: a bigger power means a bigger number (for positive numbers). $3 \times 1 0^{5} > 9 \times 1 0^{4}$ .

If the powers are equal, compare the values of $A$ .

Remember that $1 0^{- 3}$ is bigger than $1 0^{- 5}$ : $2 \times 1 0^{- 3} = 0.002$ while $8 \times 1 0^{- 5} = 0.000 08$ .

1Smallest power first:

1 0^{- 4}

, then

1 0^{- 3}

(compare 4.2 and 5), then

1 0^{- 2}

Answer

6.1 \times 1 0^{- 4}, 4.2 \times 1 0^{- 3}, 5 \times 1 0^{- 3}, 3.9 \times 1 0^{- 2}

Multiplying and dividing

Multiply the $A$ values and add the powers of 10: $(3 \times 1 0^{4}) \times (5 \times 1 0^{6}) = 15 \times 1 0^{10}$ .

Then adjust into standard form if $A$ is no longer between 1 and 10: $15 \times 1 0^{10} = 1.5 \times 1 0^{11}$ .

For division, divide the $A$ values and subtract the powers.

2.4 \div 8 = 0.3

1 0^{3} \div 1 0^{- 5} = 1 0^{8}

0.3 \times 1 0^{8} = 3 \times 1 0^{7}

Answer

3 \times 1 0^{7}

Your turnSame method, new numbers — try it before moving on.

Work out $(6 \times 1 0^{4}) \times (4 \times 1 0^{- 7})$ in standard form.

Your answer

Type it like 4.2 x 10^-3.

Why this works

When $A$ drops below 1, shift one power of 10 across: $0.3 \times 1 0^{8}$ becomes $3 \times 1 0^{7}$ . Making $A$ ten times bigger means making the power of ten ten times smaller.

Adding and subtracting

You cannot simply add the $A$ values unless the powers of 10 match. Either convert both numbers to ordinary form, or rewrite one so the powers agree.

$6.2 \times 1 0^{5} + 4.3 \times 1 0^{4} = 6.2 \times 1 0^{5} + 0.43 \times 1 0^{5} = 6.63 \times 1 0^{5}$ .

1Convert:

1260000 - 48000 = 1212000

2Write in standard form.

Answer

1.212 \times 1 0^{6}

Your turnSame method, new numbers — try it before moving on.

Work out $(3.2 \times 1 0^{5}) + (4.5 \times 1 0^{4})$ in standard form.

Your answer

Type it like 4.2 x 10^5.

Tip — On a calculator paper, use the $\times 1 0^{x}$ key and put brackets round each number. It avoids the most common input errors.

Think like an examiner

Give final answers in correct standard form, with $1 \leq A < 10$ .
Show the separate treatment of the numbers and the powers of 10 when working without a calculator.
For addition and subtraction, show the matched powers or the ordinary numbers used.

Standard form

A \times 1 0^{n}, 1 \leq A < 10

Definition.

(a \times 1 0^{m}) (b \times 1 0^{n}) = ab \times 1 0^{m + n}

Multiplying.

\frac{a \times 1 0 ^{m}}{b \times 1 0 ^{n}} = \frac{a}{b} \times 1 0^{m - n}

Dividing.

Watch out for these

Writing

0.0072

7.2 \times 1 0^{3}

Small numbers have negative powers:

7.2 \times 1 0^{- 3}

Leaving an answer as

25 \times 1 0^{6}

Adjust

A

2.5 \times 1 0^{7}

Adding

3 \times 1 0^{4} + 2 \times 1 0^{3} = 5 \times 1 0^{7}

Match powers first:

3.2 \times 1 0^{4}

Thinking

5 \times 1 0^{- 6} > 2 \times 1 0^{- 3}

because

- 6

has a bigger digit.

1 0^{- 3}

is larger than

1 0^{- 6}

Stretch yourself

A computer processes $3.6 \times 1 0^{9}$ operations per second. How many seconds would it take to process $9 \times 1 0^{15}$ operations? Give your answer in standard form, and then in days to 3 significant figures.

Hint — Divide the total by the rate. There are $86400$ seconds in a day.

Questions students ask

Why must

A

be less than 10?

It makes every number have exactly one standard form, so the power of 10 always shows the order of magnitude.

1 \times 1 0^{3}

in standard form?

Yes.

A = 1

is allowed because the condition is

1 \leq A < 10

What is

1 0^{0}

in standard form?

1 \times 1 0^{0} = 1

Key takeaways

Standard form is $A \times 1 0^{n}$ with $1 \leq A < 10$ .
Large numbers have positive $n$ ; numbers below 1 have negative $n$ .
Order by the power first, then by $A$ .
Multiply or divide the $A$ values and combine the powers with index laws, then adjust.
Match powers (or use ordinary numbers) before adding or subtracting.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 42 marks

(a)Write $7.2 \times 1 0^{- 3}$ as an ordinary number.[1]
(b)Write $58000000$ in standard form.[1]

Question 2Grade 53 marks

Light travels at $3 \times 1 0^{5}$ km per second. The Sun is $1.5 \times 1 0^{8}$ km from the Earth.

Work out how long light takes to travel from the Sun to the Earth. Give your answer in minutes and seconds.[3]

Question 3Grade 62 marks

The mass of an electron is $9.11 \times 1 0^{- 31}$ kg. Work out the mass of $5 \times 1 0^{23}$ electrons. Give your answer in standard form to 3 significant figures.[2]

Unlimited practice

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Standard Form

Writing numbers in standard form

Ordering

Multiplying and dividing

Adding and subtracting

Practise it the way it is examined

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Related topics

Standard Form

Writing numbers in standard form

Ordering

Multiplying and dividing

Adding and subtracting

Practise it the way it is examined

Skill drill

Practise this topic

Related topics