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N9NumberFoundation & HigherGrades 4–7

Standard Form

Standard form writes very large or very small numbers compactly as A×10nA \times 10^n, where 1≤A<101 \le A < 10 and nn is an integer. The distance to the Sun, about 150,000,000 km, becomes 1.5×1081.5 \times 10^8 km; a red blood cell's width, 0.000 007 m, becomes 7×10−67 \times 10^{-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
1

Writing in standard form

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

Example 1Write (a) 430,000430{,}000 (b) 0.000 6250.000\,625 in standard form.
1(a) A=4.3A = 4.3; the point moves 5 places: 4.3×1054.3 \times 10^5.
2(b) A=6.25A = 6.25; the point moves 4 places the other way: 6.25×10−46.25 \times 10^{-4}.
Answer
(a) 4.3×1054.3 \times 10^5 (b) 6.25×10−46.25 \times 10^{-4}
Your turnSame method, new numbers — try it before moving on.

Write 0.000 0380.000\,038 in standard form.

Type it like 4.2 x 10^-3.

Show worked solution
Solution
  1. 1A=3.8A = 3.8.
  2. 2The point moves 5 places right, so the power is −5-5.

Answer: 3.8×10−53.8 \times 10^{-5}

Example 2Write 7.02×1067.02 \times 10^6 as an ordinary number.
1Move the digits 6 places left (the point 6 places right).
Answer
7,020,0007{,}020{,}000
Your turnSame method, new numbers — try it before moving on.

Write 2.4×10−32.4 \times 10^{-3} as an ordinary number.

Show worked solution
Solution
  1. 1A power of −3-3 means move the point 3 places left.
  2. 22.4→0.00242.4 \to 0.0024

Answer: 0.00240.0024

Tip — 23×10423 \times 10^4 is not standard form, because 23 is not between 1 and 10. Rewrite it as 2.3×1052.3 \times 10^5.

2

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 10−310^{-3} is bigger than 10−510^{-5}.

Example 1Order from smallest: 5.1×10−35.1 \times 10^{-3}, 9.8×10−49.8 \times 10^{-4}, 2×10−32 \times 10^{-3}, 1.1×10−21.1 \times 10^{-2}.
1Smallest power: 10−410^{-4}, so 9.8×10−49.8 \times 10^{-4} first.
2Then 10−310^{-3}: 2×10−3<5.1×10−32 \times 10^{-3} < 5.1 \times 10^{-3}.
3Largest power: 1.1×10−21.1 \times 10^{-2}.
Answer
9.8×10−4, 2×10−3, 5.1×10−3, 1.1×10−29.8 \times 10^{-4}, \ 2 \times 10^{-3}, \ 5.1 \times 10^{-3}, \ 1.1 \times 10^{-2}
3

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.

Example 1Work out (6×104)×(8×107)(6 \times 10^4) \times (8 \times 10^7). Give your answer in standard form.
1Front numbers: 6×8=486 \times 8 = 48. Powers: 104+7=101110^{4+7} = 10^{11}.
248×101148 \times 10^{11} is not standard form: 48=4.8×10148 = 4.8 \times 10^1.
3So 4.8×10124.8 \times 10^{12}.
Answer
4.8×10124.8 \times 10^{12}
Your turnSame method, new numbers — try it before moving on.

Work out (5×103)×(7×10−8)(5 \times 10^{3}) \times (7 \times 10^{-8}). Give your answer in standard form.

Type it like 4.2 x 10^-3.

Show hint

5×7=355 \times 7 = 35, and 3+(−8)=−53 + (-8) = -5. Then adjust 35.

Show worked solution
Solution
  1. 135×10−535 \times 10^{-5}.
  2. 235=3.5×1035 = 3.5 \times 10, so 3.5×10−43.5 \times 10^{-4}.

Answer: 3.5×10−43.5 \times 10^{-4}

Example 2Work out 1.2×1054×10−2\dfrac{1.2 \times 10^5}{4 \times 10^{-2}}.
1Front numbers: 1.2÷4=0.31.2 \div 4 = 0.3. Powers: 105−(−2)=10710^{5 - (-2)} = 10^7.
20.3=3×10−10.3 = 3 \times 10^{-1}, so 3×1063 \times 10^6.
Answer
3×1063 \times 10^6
Your turnSame method, new numbers — try it before moving on.

Work out 9×1063×10−2\dfrac{9 \times 10^{6}}{3 \times 10^{-2}}. Give your answer in standard form.

Type it like 4 x 10^5.

Show worked solution
Solution
  1. 19÷3=39 \div 3 = 3.
  2. 2106−(−2)=10810^{6 - (-2)} = 10^8.

Answer: 3×1083 \times 10^{8}

Keeping the value the same

When you rewrite 4848 as 4.8×104.8 \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.

4

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 ×10x\times 10^x (or EXP) key. Type 3.2×10−43.2 \times 10^{-4} as 3.2, ×10x\times 10^x, −4-4.

Example 1Work out (4.6×105)+(8×104)(4.6 \times 10^5) + (8 \times 10^4). Give your answer in standard form.
1Ordinary numbers: 460,000+80,000=540,000460{,}000 + 80{,}000 = 540{,}000.
Answer
5.4×1055.4 \times 10^5
Your turnSame method, new numbers — try it before moving on.

Work out (3×104)−(6×103)(3 \times 10^{4}) - (6 \times 10^{3}). Give your answer in standard form.

Type it like 4.2 x 10^3.

Show worked solution
Solution
  1. 130,000−6,000=24,00030{,}000 - 6{,}000 = 24{,}000.
  2. 2=2.4×104= 2.4 \times 10^4.

Answer: 2.4×1042.4 \times 10^{4}

Think like an examiner

  • An answer of 48×101148 \times 10^{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 3.5×10−43.5 \times 10^{-4}.
  • In context questions (e.g. mass of atoms, population), include units and read carefully whether standard form is required.

Standard form

A×10n,1≤A<10A \times 10^n, \quad 1 \le A < 10
nn is a whole number (positive or negative).
(a×10m)(b×10n)=ab×10m+n(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}
Then adjust abab into range.
a×10mb×10n=ab×10m−n\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}
Division.

Watch out for these

0.0052=5.2×1030.0052 = 5.2 \times 10^3
Numbers less than 1 have a negative power: 5.2×10−35.2 \times 10^{-3}.
(2×103)(4×105)=8×1015(2 \times 10^3)(4 \times 10^5) = 8 \times 10^{15}
Add the powers: 8×1088 \times 10^8.
Leaving 0.3×1070.3 \times 10^7 as the answer.
0.3=3×10−10.3 = 3 \times 10^{-1}, so 3×1063 \times 10^6.
2×104+3×104=5×1082 \times 10^4 + 3 \times 10^4 = 5 \times 10^8
Adding does not add powers: 5×1045 \times 10^4.

Stretch yourself

The mass of the Earth is about 5.97×10245.97 \times 10^{24} kg. The mass of the Moon is about 7.35×10227.35 \times 10^{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.

Reveal solution
Solution5.97×10247.35×1022=5.977.35×102=0.8122…×100=81.2…\frac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = \frac{5.97}{7.35} \times 10^2 = 0.8122\ldots \times 100 = 81.2\ldots The Earth is about 81 times heavier than the Moon. ■\blacksquare

Questions students ask

Is 10×10310 \times 10^3 in standard form?
No — the front number must be less than 10. It equals 1×1041 \times 10^4.
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. 2×104+3×1042 \times 10^4 + 3 \times 10^4 is 20,000 + 30,000 = 50,000, which is 5×1045 \times 10^4.

Key takeaways

  • Standard form is A×10nA \times 10^n with 1≤A<101 \le 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
  1. (a)Write 0.000 4070.000\,407 in standard form.[1]
    Show mark scheme (a)

    Answer: 4.07×10−44.07 \times 10^{-4}

  2. (b)Write 3.9×1053.9 \times 10^5 as an ordinary number.[1]
    Show mark scheme (b)

    Answer: 390,000390{,}000

Answer on paper first, then mark it.
Question 2Grade 52 marks
  1. Work out (2.5×10−3)×(6×109)(2.5 \times 10^{-3}) \times (6 \times 10^{9}) without a calculator. Give your answer in standard form.[2]
    Show mark scheme

    Answer: 1.5×1071.5 \times 10^7

Answer on paper first, then mark it.
Question 3Grade 74 marks

A virus is 2×10−72 \times 10^{-7} m long. A bacterium is 3.2×10−63.2 \times 10^{-6} m long.

  1. (a)How many times longer is the bacterium than the virus?[2]
    Show mark scheme (a)

    Answer: 1616

  2. (b)5,000 viruses are placed end to end. Work out their total length in standard form.[2]
    Show mark scheme (b)

    Answer: 1×10−31 \times 10^{-3} m

Answer on paper first, then mark it.

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Write 52,00052{,}000 in standard form.

    • 5.2×1045.2 \times 10^4
    • 52×10352 \times 10^3
    • 5.2×1035.2 \times 10^3
    • 0.52×1050.52 \times 10^5
    Show answer and explanation

    Answer: 5.2×1045.2 \times 10^4

    The point moves 4 places.

  2. Write 0.0060.006 in standard form.

    • 6×10−36 \times 10^{-3}
    • 6×1036 \times 10^3
    • 0.6×10−20.6 \times 10^{-2}
    • 6×10−26 \times 10^{-2}
    Show answer and explanation

    Answer: 6×10−36 \times 10^{-3}

    3 places to the right.

  3. Write 8.1×10−28.1 \times 10^{-2} as an ordinary number.

    • 0.0810.081
    • 810810
    • 0.00810.0081
    • 0.810.81
    Show answer and explanation

    Answer: 0.0810.081

    Move the point 2 places left.

  4. Which is largest?

    • 2×1052 \times 10^{5}
    • 9×1049 \times 10^{4}
    • 9.9×1049.9 \times 10^{4}
    • 1×1051 \times 10^{5}
    Show answer and explanation

    Answer: 2×1052 \times 10^{5}

    Highest power, then larger front number.

  5. Work out (3×104)×(2×105)(3 \times 10^4) \times (2 \times 10^5).

    • 6×1096 \times 10^9
    • 6×10206 \times 10^{20}
    • 5×1095 \times 10^9
    • 6×1016 \times 10^1
    Show answer and explanation

    Answer: 6×1096 \times 10^9

    Multiply 3 and 2; add 4 and 5.

  6. Work out 8×1062×102\frac{8 \times 10^6}{2 \times 10^2}.

    • 4×1044 \times 10^4
    • 4×1034 \times 10^3
    • 6×1046 \times 10^4
    • 4×1084 \times 10^8
    Show answer and explanation

    Answer: 4×1044 \times 10^4

    8÷2=48 \div 2 = 4 and 6−2=46 - 2 = 4.

  7. Work out (4×103)×(5×10−6)(4 \times 10^3) \times (5 \times 10^{-6}).

    • 2×10−22 \times 10^{-2}
    • 20×10−320 \times 10^{-3}
    • 2×10−32 \times 10^{-3}
    • 9×10−39 \times 10^{-3}
    Show answer and explanation

    Answer: 2×10−22 \times 10^{-2}

    20×10−3=2×10−220 \times 10^{-3} = 2 \times 10^{-2}.

  8. Work out (6×105)+(4×104)(6 \times 10^5) + (4 \times 10^4).

    • 6.4×1056.4 \times 10^5
    • 10×10910 \times 10^9
    • 1×10101 \times 10^{10}
    • 6.04×1056.04 \times 10^5
    Show answer and explanation

    Answer: 6.4×1056.4 \times 10^5

    600,000+40,000600{,}000 + 40{,}000.

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