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 $150\,000\,000$ km; a red blood cell is about $0.000\,008$ m across. Writing out all those zeros is slow and error-prone. $\textbf{Standard form}$ writes any number as a value between 1 and 10 multiplied by a power of 10 — so those become $1.5 \times 10^{8}$ km and $8 \times 10^{-6}$ m.

45 min Powers, Roots & Standard Form

The big picture

Standard form separates a number into its digits ( $A$ ) and its size ( $10^{n}$ ). 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 $A \times 10^{n}$ where $1 \le A < 10$ and $n$ is an integer.

For large numbers, $n$ is positive: count how many places the digits move to make $A$ . $47\,000 = 4.7 \times 10^{4}$ .

For numbers less than 1, $n$ is negative: $0.000\,36 = 3.6 \times 10^{-4}$ .

Example 1Write (a)

602\,000\,000

and (b)

0.000\,050\,7

in standard form.

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 10^{8}

(b)

5.07 \times 10^{-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.

Show worked solution

Solution

1 $A = 3.09$ .
2The point moves 4 places right: $3.09 \times 10^{-4}$ .

Answer: $3.09 \times 10^{-4}$

Tip — Check $A$ first: it must be at least 1 and less than 10. $45 \times 10^{3}$ and $0.45 \times 10^{5}$ both equal $45\,000$ but neither is in standard form.

Ordering

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

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

Remember that $10^{-3}$ is bigger than $10^{-5}$ : $2 \times 10^{-3} = 0.002$ while $8 \times 10^{-5} = 0.000\,08$ .

Example 1Order from smallest:

4.2 \times 10^{-3}

3.9 \times 10^{-2}

5 \times 10^{-3}

6.1 \times 10^{-4}

1Smallest power first:

10^{-4}

, then

10^{-3}

(compare 4.2 and 5), then

10^{-2}

Answer

6.1 \times 10^{-4},\ 4.2 \times 10^{-3},\ 5 \times 10^{-3},\ 3.9 \times 10^{-2}

Multiplying and dividing

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

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

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

Example 1Without a calculator, work out

\dfrac{2.4 \times 10^{3}}{8 \times 10^{-5}}

, giving your answer in standard form.

2.4 \div 8 = 0.3

10^{3} \div 10^{-5} = 10^{8}

0.3 \times 10^{8} = 3 \times 10^{7}

Answer

3 \times 10^{7}

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

Work out $(6 \times 10^4) \times (4 \times 10^{-7})$ in standard form.

Your answer

Type it like 4.2 x 10^-3.

Show worked solution

Solution

1 $6 \times 4 = 24$ and $10^{4 + (-7)} = 10^{-3}$ .
2 $24 \times 10^{-3} = 2.4 \times 10^{-2}$ .

Answer: $2.4 \times 10^{-2}$

Why this works

When $A$ drops below 1, shift one power of 10 across: $0.3 \times 10^{8}$ becomes $3 \times 10^{7}$ . 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.

$6.2 \times 10^{5} + 4.3 \times 10^{4} = 6.2 \times 10^{5} + 0.43 \times 10^{5} = 6.63 \times 10^{5}$ .

Example 1The population of a town is

4.8 \times 10^{4}

and of a city is

1.26 \times 10^{6}

. How many more people live in the city? Give your answer in standard form.

1Convert:

1\,260\,000 - 48\,000 = 1\,212\,000

2Write in standard form.

Answer

1.212 \times 10^{6}

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

Work out $(3.2 \times 10^5) + (4.5 \times 10^4)$ in standard form.

Your answer

Type it like 4.2 x 10^5.

Show worked solution

Solution

1 $320{,}000 + 45{,}000 = 365{,}000$ .
2 $= 3.65 \times 10^5$ .

Answer: $3.65 \times 10^{5}$

Tip — On a calculator paper, use the $\times 10^{x}$ 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 $1 \le A < 10$ .
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 10^{n},\ 1 \le A < 10

Definition.

(a \times 10^{m})(b \times 10^{n}) = ab \times 10^{m+n}

Multiplying.

\tfrac{a \times 10^{m}}{b \times 10^{n}} = \tfrac{a}{b} \times 10^{m-n}

Dividing.

Watch out for these

Writing

0.0072

7.2 \times 10^{3}

Small numbers have negative powers:

7.2 \times 10^{-3}

Leaving an answer as

25 \times 10^{6}

Adjust

A

2.5 \times 10^{7}

Adding

3 \times 10^{4} + 2 \times 10^{3} = 5 \times 10^{7}

Match powers first:

3.2 \times 10^{4}

Thinking

5 \times 10^{-6} > 2 \times 10^{-3}

because

-6

has a bigger digit.

10^{-3}

is larger than

10^{-6}

Stretch yourself

A computer processes $3.6 \times 10^{9}$ operations per second. How many seconds would it take to process $9 \times 10^{15}$ operations? Give your answer in standard form, and then in days to 3 significant figures.

Hint — Divide the total by the rate. There are $86\,400$ seconds in a day.

Reveal solution

SolutionTime

= \dfrac{9 \times 10^{15}}{3.6 \times 10^{9}} = 2.5 \times 10^{6}

seconds. In days:

\dfrac{2.5 \times 10^{6}}{86\,400} = 28.935\dots

, so about

28.9

days.

\blacksquare

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 10^{3}

in standard form?

Yes.

A = 1

is allowed because the condition is

1 \le A < 10

What is

10^{0}

in standard form?

1 \times 10^{0} = 1

Key takeaways

Standard form is $A \times 10^{n}$ with $1 \le A < 10$ .
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 $7.2 \times 10^{-3}$ as an ordinary number.[1]
Show mark scheme (a)
- B1 $0.0072$
Answer: $0.0072$
(b)Write $58\,000\,000$ in standard form.[1]
Show mark scheme (b)
- B1 $5.8 \times 10^7$
Answer: $5.8 \times 10^7$

Question 2Grade 53 marks

Light travels at $3 \times 10^5$ km per second. The Sun is $1.5 \times 10^8$ 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]
Show mark scheme
- M1 $\frac{1.5 \times 10^8}{3 \times 10^5}$
- A1 $500$ seconds
- B18 minutes 20 seconds
Answer: 8 minutes 20 seconds

Question 3Grade 62 marks

The mass of an electron is $9.11 \times 10^{-31}$ kg. Work out the mass of $5 \times 10^{23}$ electrons. Give your answer in standard form to 3 significant figures.[2]
Show mark scheme
- M1 $9.11 \times 5 \times 10^{-31 + 23}$
- A1 $4.56 \times 10^{-7}$ kg
Answer: $4.56 \times 10^{-7}$ kg

Unlimited practice

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