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Maths · Number

Powers of 10 and standard form

The mass of the Earth and the width of a cell are both too awkward to write in full. Standard form writes any number as a number between 1 and 10 times a power of 10, which makes huge and tiny numbers easy to compare and calculate with.

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Last Lesson and Before

Answer each one, then check.

  1. 1

    What is \(10^3\)?

    Show answerHide answer

    \(1000\)

  2. 2

    Last lesson: what is \(10^{-2}\)?

    Show answerHide answer

    \(\frac{1}{100} = 0.01\)

  3. 3

    Work out \(4.5 \times 100\).

    Show answerHide answer

    \(450\)

  4. 4

    Write \(10^5 \times 10^3\) as a single power of 10.

    Show answerHide answer

    \(10^8\)

  5. 5

    Last lesson: what is \(5^0\)?

    Show answerHide answer

    \(1\)

Learning Objectives

  1. 1Write positive and negative powers of 10 as ordinary numbers.
  2. 2Use metric prefixes such as kilo, mega, milli and micro.
  3. 3Write numbers in standard form, and change them back.
  4. 4Multiply and divide numbers in standard form.
  5. 5Add and subtract numbers in standard form, and use a calculator with them.

Powers of 10

Multiplying by \(10^n\) moves every digit \(n\) places.

  • Positive powers

    \(10^6 = 1\,000\,000\). \(3.2 \times 10^4 = 32\,000\): the digits move 4 places left.

  • Negative powers

    \(10^{-3} = \frac{1}{1000} = 0.001\). \(3.2 \times 10^{-4} = 0.000\,32\): the digits move 4 places right.

  • Multiplying powers of 10

    \(10^5 \times 10^3 = 10^8\). Add the indices, as for any base.

  • Dividing powers of 10

    \(10^5 \div 10^8 = 10^{-3}\). Subtract the indices.

Metric Prefixes

Each prefix stands for a power of 10.

  • giga

    Symbol: G. Power of 10: \(10^9\). Example: 1 gigabyte = 1 000 000 000 bytes

  • mega

    Symbol: M. Power of 10: \(10^6\). Example: 1 megawatt = 1 000 000 watts

  • kilo

    Symbol: k. Power of 10: \(10^3\). Example: 1 kilometre = 1000 metres

  • milli

    Symbol: m. Power of 10: \(10^{-3}\). Example: 1 millimetre = 0.001 metres

  • micro

    Symbol: μ. Power of 10: \(10^{-6}\). Example: 1 micrometre = 0.000 001 metres

  • nano

    Symbol: n. Power of 10: \(10^{-9}\). Example: 1 nanosecond = 0.000 000 001 seconds

What Is Standard Form?

A number in standard form is written as \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is a whole number.

  • The first part

    A number from 1 up to (but not including) 10: one non-zero digit before the decimal point.

  • The power

    How many places the digits have moved. Positive for big numbers, negative for small numbers (less than 1).

  • Big numbers

    \(45\,000 = 4.5 \times 10^4\).

  • Small numbers

    \(0.0072 = 7.2 \times 10^{-3}\).

Writing Numbers in Standard Form

  • \(45\,000\)

    Standard form: \(4.5 \times 10^4\). How: 4.5 moved 4 places left

  • \(3\,080\,000\)

    Standard form: \(3.08 \times 10^6\). How: 3.08 moved 6 places left

  • \(0.0072\)

    Standard form: \(7.2 \times 10^{-3}\). How: 7.2 moved 3 places right

  • \(0.000\,050\,1\)

    Standard form: \(5.01 \times 10^{-5}\). How: 5.01 moved 5 places right

  • \(38 \times 10^4\)

    Standard form: \(3.8 \times 10^5\). How: \(38 = 3.8 \times 10^1\), so add 1 to the power

  • \(0.6 \times 10^{-2}\)

    Standard form: \(6 \times 10^{-3}\). How: \(0.6 = 6 \times 10^{-1}\), so subtract 1 from the power

Is It in Standard Form?

Yes

  • \(3.2 \times 10^5\)
  • \(9 \times 10^{-4}\)
  • \(1.07 \times 10^{12}\)
  • \(5 \times 10^0\), which is 5

No - and why

  • \(32 \times 10^4\): 32 is not less than 10.
  • \(0.9 \times 10^{-3}\): 0.9 is less than 1.
  • \(1.07 \times 5^{12}\): it must be a power of 10.
  • \(4.5 \times 10^{2.5}\): the power must be a whole number.

From Tiny to Huge

Standard form lets the whole universe fit on one scale.

  1. 1 An atom

    About \(1 \times 10^{-10}\) m across
  2. 2 A red blood cell

    About \(8 \times 10^{-6}\) m across
  3. 3 A person

    About \(1.7 \times 10^0\) m (1.7 m) tall
  4. 4 The Earth

    About \(1.3 \times 10^7\) m across
  5. 5 Our galaxy

    About \(1 \times 10^{21}\) m across

Multiplying in Standard Form

Work out \((3 \times 10^5) \times (6 \times 10^{-2})\). Give your answer in standard form.

Show the solutionHide the solution
  1. 1 Multiply the numbers \(3 \times 6 = 18\)
  2. 2 Multiply the powers: add the indices \(10^5 \times 10^{-2} = 10^3\)
  3. 3 Put them together \(18 \times 10^3\)
  4. 4 18 is not between 1 and 10: \(18 = 1.8 \times 10^1\) \(1.8 \times 10^1 \times 10^3\)

Answer\(1.8 \times 10^4\)

Dividing in Standard Form

Work out \((4.8 \times 10^6) \div (1.2 \times 10^{-3})\). Give your answer in standard form.

Show the solutionHide the solution
  1. 1 Divide the numbers \(4.8 \div 1.2 = 4\)
  2. 2 Divide the powers: subtract the indices \(10^6 \div 10^{-3} = 10^{6-(-3)} = 10^9\)
  3. 3 Put them together \(4 \times 10^9\)

Answer\(4 \times 10^9\)

Adding in Standard Form

Work out \((4.5 \times 10^4) + (3 \times 10^3)\). Give your answer in standard form.

Show the solutionHide the solution
  1. 1 The powers are different, so write both as ordinary numbers \(45\,000\) and \(3000\)
  2. 2 Add \(45\,000 + 3000 = 48\,000\)
  3. 3 Write the answer in standard form \(4.8 \times 10^4\)

Answer\(4.8 \times 10^4\)

Standard Form on a Calculator

Calculators have a key for entering powers of 10.

  • The key

    Look for the "×10 to the x" key (EXP on some models). To enter \(3.2 \times 10^{-5}\), press 3.2, then that key, then −5.

  • Reading the display

    A display like 1.8E4 means \(1.8 \times 10^4\). Always write the answer with \(\times 10\) and a power.

  • Brackets

    When dividing by a number in standard form, put it in brackets, or use the power-of-10 key.

  • Check

    Estimate first: is the power of 10 about right?

Case study

How Long Does Sunlight Take to Reach Us?

The Sun is about \(1.5 \times 10^8\) km from the Earth, and light travels at about \(3 \times 10^5\) km per second. Time = distance \(\div\) speed = \((1.5 \times 10^8) \div (3 \times 10^5) = 0.5 \times 10^3 = 5 \times 10^2\) seconds: about 500 seconds, or 8 minutes 20 seconds. So when you look at the Sun, you are seeing it as it was more than 8 minutes ago. Standard form turns a calculation with 17 digits into two small divisions.

\(1.5 \times 10^8\) km Distance from the Earth to the Sun
\(5 \times 10^2\) s Time for sunlight to reach us - about 8 minutes

Space and Cells

Use standard form for each question, without a calculator. (a) Write 6 000 000 000 000 in standard form. (b) A bacterium is \(2 \times 10^{-6}\) m long. How many would fit end to end along 1 cm (\(1 \times 10^{-2}\) m)? (c) Put in order, smallest first: \(3.1 \times 10^4\), \(2.9 \times 10^5\), \(31\,000\), \(0.3 \times 10^5\). (d) Work out \((2.5 \times 10^3) \times (4 \times 10^6)\).

1. Write each number in standard form first.

2. Deal with the numbers and the powers separately.

3. Check the first part is between 1 and 10.

A good answer shows: (a) \(6 \times 10^{12}\). (b) \((1 \times 10^{-2}) \div (2 \times 10^{-6}) = 0.5 \times 10^4 = 5 \times 10^3 = 5000\). (c) \(0.3 \times 10^5\) (30 000), then \(3.1 \times 10^4\) and 31 000 (equal), then \(2.9 \times 10^5\). (d) \(10 \times 10^9 = 1 \times 10^{10}\).

Can I...?

  1. 1Write powers of 10 as ordinary numbers.
  2. 2Use metric prefixes.
  3. 3Write a large number in standard form.
  4. 4Write a small number in standard form.
  5. 5Change standard form back to an ordinary number.
  6. 6Multiply and divide in standard form.
  7. 7Add and subtract in standard form.
  8. 8Use the power-of-10 key on a calculator.

Summary & Exam Focus

  • Standard form: \(A \times 10^n\) with \(1 \le A < 10\).
  • Positive powers for big numbers, negative powers for numbers less than 1.
  • Multiply or divide the numbers and the powers separately, then adjust.
  • To add or subtract, make the powers match or use ordinary numbers.

Exam focus

Work out \((5 \times 10^{-3}) \times (7 \times 10^8)\). Give your answer in standard form. (2 marks) (2 marks)

After multiplying or dividing, check the first number is between 1 and 10. \(35 \times 10^5\) is a correct value but not standard form, and loses the accuracy mark.

Key terms

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

Standard form
A number written as \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is an integer.
Power of 10
10 multiplied by itself a number of times, e.g. \(10^3 = 1000\).
Prefix
Letters in front of a unit that stand for a power of 10, e.g. kilo \(= 10^3\).
Ordinary number
A number written out in full, without powers of 10.
Integer
A whole number, positive, negative or zero.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 1 mark

    Write 0.000 604 in standard form.

    Show answerHide answer

    Model answer

    \(6.04 \times 10^{-4}\)

    Mark scheme

    • \(6.04 \times 10^{-4}\) — B1
  2. Question 2 Non-calculator 1 mark

    Write \(7.3 \times 10^5\) as an ordinary number.

    Show answerHide answer

    Model answer

    \(730\,000\)

    Mark scheme

    • \(730\,000\) — B1
  3. Question 3 Non-calculator 2 marks

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

    Show answerHide answer

    Model answer

    \(5 \times 7 = 35\) and \(10^{-3} \times 10^8 = 10^5\), so \(35 \times 10^5 = 3.5 \times 10^6\).

    Mark scheme

    • \(35 \times 10^5\) or \(3\,500\,000\) seen — M1
    • \(3.5 \times 10^6\) — A1
  4. Question 4 Non-calculator 2 marks

    Work out \((8.4 \times 10^7) \div (2.1 \times 10^{-2})\). Give your answer in standard form.

    Show answerHide answer

    Model answer

    \(8.4 \div 2.1 = 4\) and \(10^7 \div 10^{-2} = 10^9\), so \(4 \times 10^9\).

    Mark scheme

    • 4 seen, or \(10^9\) seen — M1
    • \(4 \times 10^9\) — A1
  5. Question 5 Non-calculator 2 marks

    Write these numbers in order of size, starting with the smallest. \(4.1 \times 10^{-3}\), \(0.0039\), \(42 \times 10^{-4}\), \(0.4 \times 10^{-2}\)

    Show answerHide answer

    Model answer

    As ordinary numbers: 0.0041, 0.0039, 0.0042, 0.004. Order: \(0.0039\), \(0.4 \times 10^{-2}\), \(4.1 \times 10^{-3}\), \(42 \times 10^{-4}\).

    Mark scheme

    • All four in the correct order — B2
    • At least three converted correctly to the same form, or one pair out of order — B1
  6. Question 6 Calculator 3 marks

    The mass of one atom of gold is \(3.27 \times 10^{-22}\) grams. How many atoms of gold are there in 1 kilogram of gold? Give your answer in standard form, correct to 3 significant figures.

    Show answerHide answer

    Model answer

    1 kg = 1000 g. \(1000 \div (3.27 \times 10^{-22}) = 3.058\ldots \times 10^{24}\), so about \(3.06 \times 10^{24}\) atoms.

    Mark scheme

    • 1 kg = 1000 g used — P1
    • \(1000 \div (3.27 \times 10^{-22})\) — P1
    • \(3.06 \times 10^{24}\) — A1

Quick check

  1. Which is 250 000 written in standard form?

    1. A\(25 \times 10^4\)
    2. B\(2.5 \times 10^4\)
    3. C\(2.5 \times 10^5\)
    4. D\(0.25 \times 10^6\)
    Show answerHide answer

    C: \(2.5 \times 10^5\)

    2.5 moves 5 places to make 250 000, so \(2.5 \times 10^5\).

  2. Which of these is NOT in standard form?

    1. A\(1.2 \times 10^4\)
    2. B\(12 \times 10^3\)
    3. C\(9.9 \times 10^{-2}\)
    4. D\(1 \times 10^6\)
    Show answerHide answer

    B: \(12 \times 10^3\)

    12 is not less than 10, so \(12 \times 10^3\) is not in standard form (it is \(1.2 \times 10^4\)).

  3. What is \((2 \times 10^3) \times (4 \times 10^5)\)?

    1. A\(8 \times 10^8\)
    2. B\(8 \times 10^{15}\)
    3. C\(6 \times 10^8\)
    4. D\(8 \times 10^2\)
    Show answerHide answer

    A: \(8 \times 10^8\)

    \(2 \times 4 = 8\) and \(10^3 \times 10^5 = 10^8\).

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