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Maths · Standard Form and Accuracy

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Calculating with Standard Form

Multiplying, dividing, adding and subtracting numbers in standard form, and solving problems in context.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Multiply and divide numbers in standard form without a calculator.
  2. 2Add and subtract numbers in standard form.
  3. 3Give the answer in standard form, adjusting it when needed.
  4. 4Solve problems in context using standard form.

Using the laws of indices

Calculations in standard form work by dealing with the numbers and the powers of 10 separately. For multiplication and division you use the laws of indices on the powers, and for addition and subtraction you write the numbers with the same power first. The answer must always be in standard form, so check whether the first part is between 1 and 10 at the end. These questions are common on the non-calculator paper, where the numbers are chosen so that the arithmetic is easy.

Multiplying and dividing

Multiply or divide the first parts, and add or subtract the powers.

  • Multiply

    \((2 \times 10^3) \times (3 \times 10^4) = 6 \times 10^{3 + 4} = 6 \times 10^7\).

  • Divide

    \((6 \times 10^8) \div (3 \times 10^3) = 2 \times 10^{8 - 3} = 2 \times 10^5\).

  • Adjust

    If the first part is 10 or more, such as \(20 \times 10^8\), write \(2 \times 10^9\).

  • Negative powers

    \((4 \times 10^{-3}) \times (2 \times 10^{-2}) = 8 \times 10^{-5}\).

Multiplying with an adjustment

Work out \((4 \times 10^5) \times (5 \times 10^3)\). Give your answer in standard form.

Show the solutionHide the solution
  1. 1 Multiply the numbers \(4 \times 5 = 20\).
  2. 2 Add the powers \(10^5 \times 10^3 = 10^8\).
  3. 3 Combine \(20 \times 10^8\), which is not in standard form.
  4. 4 Adjust \(20 \times 10^8 = 2 \times 10^9\).

Answer\(2 \times 10^9\)

Adding and subtracting

Write both numbers with the same power of 10, then add or subtract the first parts.

  • Same power

    \(3 \times 10^4 + 5 \times 10^3 = 3 \times 10^4 + 0.5 \times 10^4\).

  • Add

    \(3.5 \times 10^4\).

  • Or use ordinary numbers

    \(30\,000 + 5000 = 35\,000 = 3.5 \times 10^4\).

  • Never add the powers

    The powers are only added when multiplying.

Subtracting in standard form

Work out \(6.2 \times 10^5 - 3 \times 10^4\). Give your answer in standard form.

Show the solutionHide the solution
  1. 1 Same power \(3 \times 10^4 = 0.3 \times 10^5\).
  2. 2 Subtract \(6.2 \times 10^5 - 0.3 \times 10^5 = 5.9 \times 10^5\).
  3. 3 Check \(620\,000 - 30\,000 = 590\,000\).
  4. 4 Standard form \(590\,000 = 5.9 \times 10^5\).

Answer\(5.9 \times 10^5\)

Problems in context

Read the units, and say what the answer means.

  • Total

    Multiply the number of items by the size of each: \(5 \times 10^8\) bacteria of length \(2 \times 10^{-6}\) m is \(10 \times 10^2 = 1 \times 10^3\) m long.

  • Rate

    Divide a total by the number of items.

  • Units

    Keep the units the same throughout, and give them in the answer.

  • Reasonable

    Check the power is sensible for the situation.

Test yourself

  1. 1

    What is \((2 \times 10^3) \times (3 \times 10^4)\)?

    Show answerHide answer

    \(6 \times 10^7\).

  2. 2

    What is \((6 \times 10^8) \div (3 \times 10^3)\)?

    Show answerHide answer

    \(2 \times 10^5\).

  3. 3

    How do you add numbers in standard form?

    Show answerHide answer

    Write them with the same power, then add the first parts.

  4. 4

    What do you do with the powers when multiplying?

    Show answerHide answer

    Add them.

  5. 5

    What do you do with the powers when dividing?

    Show answerHide answer

    Subtract them.

Exam technique: calculations in standard form

Do the numbers and the powers separately.

  • Write out the steps

    Numbers first, then powers.

  • Check the first part

    If it is 10 or more, or less than 1, adjust it.

  • Use ordinary numbers to check

    Especially for addition and subtraction.

  • Give the answer in standard form

    The question usually says so.

Summary and exam focus

  • To multiply, multiply the first parts and add the powers.
  • To divide, divide the first parts and subtract the powers.
  • To add or subtract, write the numbers with the same power first.
  • Always give the final answer in standard form.

Exam focus

Work out \((3.6 \times 10^7) \div (1.2 \times 10^{-2})\). Give your answer in standard form. (2 marks) (2 marks)

\(3.6 \div 1.2 = 3\), and \(10^7 \div 10^{-2} = 10^{7 - (-2)} = 10^9\), so the answer is \(3 \times 10^9\). Be careful with the double negative when subtracting the powers.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Standard form
A number written as \(A \times 10^n\), with \(A\) at least 1 and less than 10.
Law of indices
A rule for combining powers, such as adding the powers when multiplying.
Power
The small number showing how many times a number is multiplied by itself.
First part
The number between 1 and 10 in standard form.
Adjust
To change the first part and the power so the answer is in standard form.
Order of magnitude
The size of a number, shown by its power of 10.
Unit
The measure that goes with a number, such as metres.
Rate
A quantity per unit of another, such as a speed.
Total
The result of adding or multiplying all the parts.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 2 marks Easier

Work out \((3 \times 10^2) \times (2 \times 10^5)\). Give your answer in standard form. [2 marks]

Mark scheme — 2 marks available

  • \(3 \times 2\) or \(10^7\) — M1
  • \(6 \times 10^7\) — A1

Model answer

\(3 \times 2 = 6\) and \(10^2 \times 10^5 = 10^7\), so \(6 \times 10^7\).

2. Exam question Work out 2 marks Easier

Work out \((8 \times 10^9) \div (4 \times 10^6)\). Give your answer in standard form. [2 marks]

Mark scheme — 2 marks available

  • \(8 \div 4\) or \(10^3\) — M1
  • \(2 \times 10^3\) — A1

Model answer

\(8 \div 4 = 2\) and \(10^9 \div 10^6 = 10^3\), so \(2 \times 10^3\).

3. Exam question Work out 3 marks Core

Work out \((5 \times 10^4) \times (6 \times 10^3)\). Give your answer in standard form. [3 marks]

Mark scheme — 3 marks available

  • \(30 \times 10^7\) — M1
  • Adjusts to standard form — M1
  • \(3 \times 10^8\) — A1

Model answer

\(5 \times 6 = 30\) and \(10^4 \times 10^3 = 10^7\), giving \(30 \times 10^7 = 3 \times 10^8\).

4. Exam question Work out 3 marks Core

Work out \(4 \times 10^5 + 6 \times 10^4\). Give your answer in standard form. [3 marks]

Mark scheme — 3 marks available

  • Same power or ordinary numbers — M1
  • 460 000 or \(4.6 \times 10^5\) seen — M1
  • \(4.6 \times 10^5\) — A1

Model answer

\(400\,000 + 60\,000 = 460\,000 = 4.6 \times 10^5\).

5. Exam question Work out 3 marks Stretch

A grain of sand is \(2 \times 10^{-3}\) m wide. \(5 \times 10^6\) grains are put in a line, side by side. Work out the length of the line. Give your answer in standard form. [3 marks]

Mark scheme — 3 marks available

  • \(5 \times 2 = 10\) or \(10^6 \times 10^{-3} = 10^3\) — M1
  • \(10 \times 10^3\) — M1
  • \(1 \times 10^4\) m — A1

Model answer

\((5 \times 10^6) \times (2 \times 10^{-3}) = 10 \times 10^3 = 1 \times 10^4\) m.

6. Exam question Work out 3 marks Stretch

Work out \((9 \times 10^{-3}) \div (3 \times 10^{-5})\). Give your answer in standard form. [3 marks]

Mark scheme — 3 marks available

  • \(9 \div 3 = 3\) — M1
  • \(10^2\) seen — M1
  • \(3 \times 10^2\) — A1

Model answer

\(9 \div 3 = 3\) and \(10^{-3} \div 10^{-5} = 10^{-3 - (-5)} = 10^2\), so the answer is \(3 \times 10^2\).

7. Multiple choice 1 mark Easier

What is \((2 \times 10^3) \times (3 \times 10^4)\)?

  1. A \(6 \times 10^{12}\)
  2. B \(5 \times 10^7\)
  3. C \(6 \times 10^7\) Correct
  4. D \(6 \times 10^1\)

Why: Multiply \(2 \times 3 = 6\) and add the powers, \(3 + 4 = 7\).

8. Multiple choice 1 mark Easier

What is \((6 \times 10^8) \div (3 \times 10^3)\)?

  1. A \(2 \times 10^{11}\)
  2. B \(2 \times 10^5\) Correct
  3. C \(3 \times 10^5\)
  4. D \(2 \times 10^{8}\)

Why: Divide \(6 \div 3 = 2\) and subtract the powers, \(8 - 3 = 5\).

9. Multiple choice 1 mark Core

What is \((4 \times 10^5) \times (5 \times 10^3)\) in standard form?

  1. A \(2 \times 10^9\) Correct
  2. B \(20 \times 10^8\)
  3. C \(2 \times 10^8\)
  4. D \(9 \times 10^8\)

Why: \(4 \times 5 = 20\) and \(10^5 \times 10^3 = 10^8\), so \(20 \times 10^8 = 2 \times 10^9\).

10. Multiple choice 1 mark Core

What is \(3 \times 10^4 + 5 \times 10^3\) in standard form?

  1. A \(8 \times 10^4\)
  2. B \(8 \times 10^7\)
  3. C \(3.5 \times 10^3\)
  4. D \(3.5 \times 10^4\) Correct

Why: \(30\,000 + 5000 = 35\,000 = 3.5 \times 10^4\).

11. Multiple choice 1 mark Core

What is \(6.2 \times 10^5 - 3 \times 10^4\) in standard form?

  1. A \(3.2 \times 10^1\)
  2. B \(5.9 \times 10^4\)
  3. C \(5.9 \times 10^5\) Correct
  4. D \(3.2 \times 10^5\)

Why: \(620\,000 - 30\,000 = 590\,000 = 5.9 \times 10^5\).

12. Multiple choice 1 mark Core

What is \(10^7 \div 10^{-2}\)?

  1. A \(10^5\)
  2. B \(10^9\) Correct
  3. C \(10^{-5}\)
  4. D \(10^{-9}\)

Why: Subtract the powers: \(7 - (-2) = 9\).

13. Multiple choice 1 mark Core

What is \((4 \times 10^{-3}) \times (2 \times 10^{-2})\)?

  1. A \(8 \times 10^{-5}\) Correct
  2. B \(8 \times 10^{-6}\)
  3. C \(6 \times 10^{-5}\)
  4. D \(8 \times 10^{5}\)

Why: Multiply \(4 \times 2 = 8\) and add the powers, \(-3 + (-2) = -5\).

14. Multiple choice 1 mark Stretch

What is \((3.6 \times 10^7) \div (1.2 \times 10^{-2})\)?

  1. A \(3 \times 10^5\)
  2. B \(3 \times 10^{-9}\)
  3. C \(4.8 \times 10^5\)
  4. D \(3 \times 10^9\) Correct

Why: \(3.6 \div 1.2 = 3\) and \(10^7 \div 10^{-2} = 10^9\).

15. Multiple choice 1 mark Stretch

\(5 \times 10^8\) bacteria, each \(2 \times 10^{-6}\) m long, are placed end to end. How long is the line?

  1. A \(1 \times 10^2\) m
  2. B \(10 \times 10^2\) m
  3. C \(1 \times 10^3\) m Correct
  4. D \(1 \times 10^{14}\) m

Why: \(5 \times 2 = 10\) and \(10^8 \times 10^{-6} = 10^2\), so \(10 \times 10^2 = 1 \times 10^3\).