Is 10 10^3 in standard form?

No — the front number must be less than 10. It equals 1 10^4.

N9NumberFoundation & HigherGrades 4–7

Standard Form

Q: How do I know whether the power is positive or negative?

If the ordinary number is 10 or bigger, the power is positive. If it is between 0 and 1, the power is negative. Numbers from 1 to under 10 have power 0.

Q: Why can I not just add the powers when adding?

Adding powers corresponds to multiplying. 2 10^4 + 3 10^4 is 20,000 + 30,000 = 50,000, which is 5 10^4.

Standard form writes very large or very small numbers compactly as $A \times 1 0^{n}$ , where $1 \leq A < 10$ and $n$ is an integer. The distance to the Sun, about 150,000,000 km, becomes $1.5 \times 1 0^{8}$ km; a red blood cell's width, 0.000 007 m, becomes $7 \times 1 0^{- 6}$ m.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

The power of 10 records how many places the digits have moved — positive for big numbers, negative for small ones. It is really just a precise way of stating the size of a number, which is why scientists use it.

Calculations become index-law problems: multiply the front numbers, add the powers, then adjust so the front number is back between 1 and 10. On the non-calculator paper, that final adjustment is where most marks are lost.

Standard FormCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Write numbers in standard form and convert back to ordinary numbers
Order numbers written in standard form
Multiply and divide in standard form without a calculator
Add and subtract in standard form, and use a calculator for standard form

Writing in standard form

Place the decimal point after the first non-zero digit to get $A$ . Count how many places the point moved: that is $n$ . Large numbers (10 or more) have a positive power; numbers less than 1 have a negative power.

1(a)

A = 4.3

; the point moves 5 places:

4.3 \times 1 0^{5}

2(b)

A = 6.25

; the point moves 4 places the other way:

6.25 \times 1 0^{- 4}

Answer(a)

4.3 \times 1 0^{5}

(b)

6.25 \times 1 0^{- 4}

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

Write $0.000 038$ in standard form.

Your answer

Type it like 4.2 x 10^-3.

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

Write $2.4 \times 1 0^{- 3}$ as an ordinary number.

Your answer

Tip — $23 \times 1 0^{4}$ is not standard form, because 23 is not between 1 and 10. Rewrite it as $2.3 \times 1 0^{5}$ .

Ordering

Compare the powers of 10 first — a bigger power means a bigger number. Only when the powers are equal do you compare the front numbers. Remember $1 0^{- 3}$ is bigger than $1 0^{- 5}$ .

1Smallest power:

1 0^{- 4}

, so

9.8 \times 1 0^{- 4}

first.

2Then

1 0^{- 3}

2 \times 1 0^{- 3} < 5.1 \times 1 0^{- 3}

3Largest power:

1.1 \times 1 0^{- 2}

Answer

9.8 \times 1 0^{- 4}, 2 \times 1 0^{- 3}, 5.1 \times 1 0^{- 3}, 1.1 \times 1 0^{- 2}

Multiplying and dividing

Multiply (or divide) the front numbers, and add (or subtract) the powers using the index laws. If the front number is no longer between 1 and 10, adjust it and change the power to compensate.

1Front numbers:

6 \times 8 = 48

. Powers:

1 0^{4 + 7} = 1 0^{11}

48 \times 1 0^{11}

is not standard form:

48 = 4.8 \times 1 0^{1}

3So

4.8 \times 1 0^{12}

Answer

4.8 \times 1 0^{12}

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

Work out $(5 \times 1 0^{3}) \times (7 \times 1 0^{- 8})$ . Give your answer in standard form.

Your answer

Type it like 4.2 x 10^-3.

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

Work out $\frac{9 \times 1 0 ^{6}}{3 \times 1 0 ^{- 2}}$ . Give your answer in standard form.

Your answer

Type it like 4 x 10^5.

Keeping the value the same

When you rewrite $48$ as $4.8 \times 10$ , you divide the front by 10, so you must multiply the power part by 10 to keep the value the same. Every adjustment is a trade: front down, power up — or front up, power down.

Adding, subtracting and calculators

To add or subtract, convert to ordinary numbers (or to the same power of 10) first, then convert the answer back.

On a calculator, use the $\times 1 0^{x}$ (or EXP) key. Type $3.2 \times 1 0^{- 4}$ as 3.2, $\times 1 0^{x}$ , $- 4$ .

1Ordinary numbers:

460, 000 + 80, 000 = 540, 000

Answer

5.4 \times 1 0^{5}

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

Work out $(3 \times 1 0^{4}) - (6 \times 1 0^{3})$ . Give your answer in standard form.

Your answer

Type it like 4.2 x 10^3.

Think like an examiner

An answer of $48 \times 1 0^{11}$ usually earns the method mark but not the accuracy mark — finish the adjustment.
Calculator displays such as "3.5E-4" are not acceptable final answers; write $3.5 \times 1 0^{- 4}$ .
In context questions (e.g. mass of atoms, population), include units and read carefully whether standard form is required.

Standard form

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

n

is a whole number (positive or negative).

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

Then adjust

ab

into range.

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

Division.

Watch out for these

0.0052 = 5.2 \times 1 0^{3}

Numbers less than 1 have a negative power:

5.2 \times 1 0^{- 3}

(2 \times 1 0^{3}) (4 \times 1 0^{5}) = 8 \times 1 0^{15}

Add the powers:

8 \times 1 0^{8}

Leaving

0.3 \times 1 0^{7}

as the answer.

0.3 = 3 \times 1 0^{- 1}

, so

3 \times 1 0^{6}

2 \times 1 0^{4} + 3 \times 1 0^{4} = 5 \times 1 0^{8}

Adding does not add powers:

5 \times 1 0^{4}

Stretch yourself

The mass of the Earth is about $5.97 \times 1 0^{24}$ kg. The mass of the Moon is about $7.35 \times 1 0^{22}$ kg. How many times heavier is the Earth than the Moon? Give your answer to the nearest whole number.

Hint — Divide the front numbers, then subtract the powers.

Questions students ask

10 \times 1 0^{3}

in standard form?

No — the front number must be less than 10. It equals

1 \times 1 0^{4}

How do I know whether the power is positive or negative?

If the ordinary number is 10 or bigger, the power is positive. If it is between 0 and 1, the power is negative. Numbers from 1 to under 10 have power 0.

Why can I not just add the powers when adding?

Adding powers corresponds to multiplying.

2 \times 1 0^{4} + 3 \times 1 0^{4}

is 20,000 + 30,000 = 50,000, which is

5 \times 1 0^{4}

Key takeaways

Standard form is $A \times 1 0^{n}$ with $1 \leq A < 10$ .
Big numbers have positive powers; numbers below 1 have negative powers.
Order by comparing powers first, then front numbers.
Multiply/divide: combine front numbers and use index laws, then adjust.
Add/subtract: convert to ordinary numbers or matching powers first.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 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 $0.000 407$ in standard form.[1]
(b)Write $3.9 \times 1 0^{5}$ as an ordinary number.[1]

Question 2Grade 52 marks

Work out $(2.5 \times 1 0^{- 3}) \times (6 \times 1 0^{9})$ without a calculator. Give your answer in standard form.[2]

Question 3Grade 74 marks

A virus is $2 \times 1 0^{- 7}$ m long. A bacterium is $3.2 \times 1 0^{- 6}$ m long.

(a)How many times longer is the bacterium than the virus?[2]
(b)5,000 viruses are placed end to end. Work out their total length in standard form.[2]

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Standard Form

Writing in standard form

Ordering

Multiplying and dividing

Adding, subtracting and calculators

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Related topics

Standard Form

Writing in standard form

Ordering

Multiplying and dividing

Adding, subtracting and calculators

Practise it the way it is examined

Skill drill

Practise this topic

Related topics