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GCSE Maths · Number

Standard Form GCSE Questions, Worked Examples and Answers

Standard form writes a number as a×10na\times10^n, where 1≤a<101\le a<10 and nn is an integer. It makes very large and very small values easier to read, compare and calculate with. This guide teaches conversions, all four operations, calculator notation and contextual GCSE standard form questions.

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One number, written at a useful scale

What you need to know about standard form

Standard form is a shorter way to write very large or very small numbers. It separates the useful digits from the power of 10 that gives them their place value.

46 000=4.6×10 000=4.6×10446\,000=4.6\times10\,000=4.6\times10^4

The value has not changed. The number 4.64.6 holds the significant digits, while 104=10 00010^4=10\,000 rebuilds their scale.

a×10n1≤a<10\boxed{a\times10^n}\qquad 1\le a<10

In this notation, aa is at least 1 but less than 10. The index nn is an integer — a positive or negative whole number, or zero — and tells you which power of 10 multiplies it.

In many countries this notation is called scientific notation. In UK GCSE Maths, standard form is the usual term.

Build the scale

What positive, zero and negative powers of 10 mean

Powers of 10 from 3 to −3
PowerValueWhat it means
10310^31,000multiply by 1,000
10210^2100multiply by 100
10110^110multiply by 10
10010^01the value is unchanged
10−110^{-1}0.1divide by 10
10−210^{-2}0.01divide by 100
10−310^{-3}0.001divide by 1,000

Read down the table: each time the index decreases by 1, the value is divided by 10. Read upwards and the value is multiplied by 10.

Why large numbers use positive powers

4.6×104=4.6×10 000=46 0004.6\times10^4=4.6\times10\,000=46\,000

The coefficient is multiplied by a power greater than 1. Each additional factor of 10 shifts every digit one place to a larger place value.

Why small numbers use negative powers

4.6×10−4=4.6×110 000=0.000464.6\times10^{-4}=4.6\times\frac1{10\,000}=0.00046

A negative index does not make the value negative. It means divide by a positive power of 10, producing a positive number between 0 and 1.

Think in multiplication and division

For 6.31×1056.31\times10^5, multiplying by 10510^5 makes the ordinary number 631 000631\,000. For 6.31×10−56.31\times10^{-5}, multiplying by 10−5=1÷10510^{-5}=1\div10^5 makes 0.00006310.0000631.

The decimal point appears to move because every digit changes place value. Describe the calculation as multiplying or dividing by a power of 10; this keeps the direction and the sign of the index connected to the value.

Speed check: once the place-value idea is secure, count how many factors of 10 rebuild the original number. That count gives the size of the index; the calculation above tells you whether its sign should be positive or negative.

Test checklist

What you need to be able to do with standard form

If you can do each item below, you have covered the standard-form skills that GCSE questions can test. The worked examples show them in action.

  • Recognise correct standard formCheck that the coefficient is at least 1 but less than 10, and that the power of 10 has an integer index.
  • Write ordinary numbers in standard formDo this for both large whole numbers and small decimals, choosing the correct sign for the index.
  • Convert standard form back to an ordinary numberUse the power of 10 to decide whether the coefficient becomes larger or smaller.
  • Compare and order valuesCompare the powers first; compare coefficients only when the powers are the same.
  • Multiply and divideCalculate with the coefficients, apply the index laws, then check that the final coefficient is between 1 and 10.
  • Add and subtractRewrite the numbers with the same power of 10 before combining their coefficients.
  • Use a calculator confidentlyEnter powers correctly and read displays such as 6.2E−4 as 6.2 × 10⁻⁴.
  • Solve problems with units and scaleKeep the requested units and use an estimate to check that the order of magnitude is sensible.
Fully worked

Standard form GCSE worked examples

These examples move from place value to operations, normalisation and reasoning about scale.

Example 1

Write a large number in standard form

1 mark
Question

Write 4 830 0004\,830\,000 in standard form.

The final coefficient must be between 1 and 10. Use the same significant digits to make 4.83, then find the power of 10 that rebuilds the original number.

4 830 000=4.83×1 000 0004\,830\,000=4.83\times1\,000\,000
1 000 000=1061\,000\,000=10^6
4.83×106\boxed{4.83\times10^6}

Check: multiplying 4.834.83 by 10610^6 rebuilds 4 830 0004\,830\,000. The positive index fits a large number.

Example 2

Write a small decimal in standard form

1 mark
Question

Write 0.00007260.0000726 in standard form.

The number is between 0 and 1, so its standard-form index will be negative. Keep the significant digits as 7.26 and identify the division by a power of 10.

10−5=1100 000=0.0000110^{-5}=\frac1{100\,000}=0.00001
7.26×0.00001=0.00007267.26\times0.00001=0.0000726
7.26×10−5\boxed{7.26\times10^{-5}}

Exam tip: the index is negative because the coefficient must be divided by 10510^5 to produce the small decimal. It is not negative because of how far you moved your pencil.

Example 3

Convert standard form to ordinary numbers

2 marks
Question

Write (a) 5.04×1045.04\times10^4 and (b) 3.8×10−63.8\times10^{-6} as ordinary numbers.

Use the index to read the scale: a positive index multiplies by a power of 10, while a negative index divides by one.

5.04×104=5.04×10 000=50 4005.04\times10^4=5.04\times10\,000=\boxed{50\,400}
3.8×10−6=3.8×11 000 000=0.00000383.8\times10^{-6}=3.8\times\frac1{1\,000\,000}=\boxed{0.0000038}

Check: the positive-power result is greater than 10; the negative-power result is between 0 and 1.

Example 4

Compare and order by power first

2 marks
Question

Write 6.2×10−36.2\times10^{-3}, 8.9×10−48.9\times10^{-4}, 1.03×10−21.03\times10^{-2} and 4.7×10−34.7\times10^{-3} in ascending order.

Compare the powers first because they show the scale of each number. Only compare coefficients when two numbers have the same power.

10−4<10−3<10−210^{-4}<10^{-3}<10^{-2}
4.7×10−3<6.2×10−3because 4.7<6.24.7\times10^{-3}<6.2\times10^{-3}\quad\text{because }4.7<6.2
8.9×10−4, 4.7×10−3, 6.2×10−3, 1.03×10−2\boxed{8.9\times10^{-4},\ 4.7\times10^{-3},\ 6.2\times10^{-3},\ 1.03\times10^{-2}}

Exam tip: among negative indices, −4-4 represents a smaller scale than −3-3.

Example 5

Multiply, then normalise

3 marks
Question

Work out (7.5×104)(2.8×10−6)(7.5\times10^4)(2.8\times10^{-6}). Give your answer in standard form.

Multiply the coefficients and add the indices. The first result may need adjusting so its coefficient is between 1 and 10.

7.5×2.8=217.5\times2.8=21
104×10−6=104+(−6)=10−210^4\times10^{-6}=10^{4+(-6)}=10^{-2}
21×10−221\times10^{-2}
21×10−2=2.1×10−121\times10^{-2}=2.1\times10^{-1}
2.1×10−1\boxed{2.1\times10^{-1}}

Reasonableness check: (8×104)(3×10−6)=24×10−2=2.4×10−1(8\times10^4)(3\times10^{-6})=24\times10^{-2}=2.4\times10^{-1}, close to the exact answer. Always check 1≤a<101\le a<10 after multiplying.

Example 6

Divide with negative indices

3 marks
Question

Work out 3.6×10−79×102\dfrac{3.6\times10^{-7}}{9\times10^2}. Give your answer in standard form.

Divide the coefficients and subtract the indices. Then check whether the coefficient is valid standard form.

3.6÷9=0.43.6\div9=0.4
10−7÷102=10−7−2=10−910^{-7}\div10^2=10^{-7-2}=10^{-9}
0.4×10−90.4\times10^{-9}
0.4×10−9=4×10−100.4\times10^{-9}=4\times10^{-10}
4×10−10\boxed{4\times10^{-10}}

Negative-index check: subtracting 22 from −7-7 gives −9-9. Normalising 0.40.4 then reduces the index once more.

Example 7

Subtract by matching powers

3 marks
Question

Work out 6.4×105−8.75×1046.4\times10^5-8.75\times10^4. Give your answer in standard form.

The powers must match before the coefficients can be subtracted, because the coefficients must count the same place-value units.

8.75×104=0.875×1058.75\times10^4=0.875\times10^5
6.4×105−0.875×1056.4\times10^5-0.875\times10^5
=(6.4−0.875)×105=(6.4-0.875)\times10^5
=5.525×105=5.525\times10^5
5.525×105\boxed{5.525\times10^5}

Why powers must match: 6.46.4 counts units of 10510^5, while 8.758.75 initially counts units of 10410^4. Adding or subtracting those coefficients directly would mix different place values.

Example 8

Read a calculator result

2 marks
Question

A calculator gives 1.68E−2 for (2.8×10−7)(6×104)(2.8\times10^{-7})(6\times10^4). Write the display as an ordinary number and check its order of magnitude.

Read E−2 as “multiply by 10 to the power −2”. An estimate then checks whether the calculator result has a sensible scale.

1.68E-2=1.68×10−2\text{1.68E-2}=1.68\times10^{-2}
1.68×10−2=0.01681.68\times10^{-2}=\boxed{0.0168}
(3×10−7)(6×104)=18×10−3=1.8×10−2(3\times10^{-7})(6\times10^4)=18\times10^{-3}=1.8\times10^{-2}

Calculator risk: E−2 means “multiply by 10−210^{-2}”. Do not read it as 1.68−21.68-2.

Example 9 · Context

Keep the unit and check the scale

4 marks
Question

A bacterium is 2.4×10−62.4\times10^{-6} m long. A chain contains 4.5×1054.5\times10^5 bacteria placed end to end. Find the chain length in metres.

The context asks for total length, so multiply one length by the number of bacteria. Keep the metre unit and estimate the scale.

length=(2.4×10−6)(4.5×105) m\text{length}=(2.4\times10^{-6})(4.5\times10^5)\text{ m}
=10.8×10−1 m=10.8\times10^{-1}\text{ m}
=1.08×100 m=1.08\times10^0\text{ m}
1.08 m\boxed{1.08\text{ m}}
(2×10−6)(5×105)=10×10−1=1 m(2\times10^{-6})(5\times10^5)=10\times10^{-1}=1\text{ m}

Reasonableness check: the estimate is 11 m, so 1.081.08 m has a sensible order of magnitude. The metre unit remains because a length was multiplied by a count.

Calculator notation

Using a scientific calculator with standard form

The exact button label differs by model, but the mathematical meaning is the same.

×10ˣEXPEE

Enter one standard-form number

To enter 3.7×10−53.7\times10^{-5}, type 3.7, press the standard-form key, then enter −5. Do not add another multiplication sign if the key already inserts ×10x\times10^x.

4.26E−7

Interpret the display

Read this as 4.26×10−74.26\times10^{-7}. The E is part of the display convention. It is not the constant ee and it does not subtract 7.

Norm ↔ Sci

Write the requested form

A calculator setting may change how the same value is displayed. Copying the screen is not enough: if the question says “give your answer in standard form”, make sure the written coefficient satisfies 1≤a<101\le a<10.

Non-calculator papers: you must be able to multiply and divide standard-form numbers by hand using coefficient calculations and index laws, then normalise the result. Examples 5 and 6 show the full method.

15 original questions · total 33 marks

Standard form GCSE exam questions

Attempt each question before opening its worked answer. The set covers conversion, ordering, operations, calculator notation, contexts and units.

Before you startAllow about 40 minutes · show powers and normalisation · answers start collapsed

Write a large number in standard form

1 mark

Write 7 460 0007\,460\,000 in standard form.

Show worked answer

The coefficient must start with 77 and lie between 11 and 1010.

7 460 000=7.46×1 000 0007\,460\,000=7.46\times1\,000\,000 1 000 000=1061\,000\,000=10^6 7.46×106\boxed{7.46\times10^6}

The positive index is sensible because the original number is greater than 1010.

Write a small number in standard form

1 mark

Write 0.0000005820.000000582 in standard form.

Show worked answer

Use the significant digits to form the coefficient 5.825.82.

10−7=1107=0.000000110^{-7}=\frac{1}{10^7}=0.0000001 5.82×10−7=0.0000005825.82\times10^{-7}=0.000000582 5.82×10−7\boxed{5.82\times10^{-7}}

The negative index is required because multiplying 5.825.82 by 10−710^{-7} divides it by 10710^7.

Write a large ordinary number

1 mark

Write 3.07×1053.07\times10^5 as an ordinary number.

Show worked answer

A positive index means multiply by 100 000100\,000.

3.07×105=3.07×100 0003.07\times10^5=3.07\times100\,000 307 000\boxed{307\,000}

Write a small ordinary number

1 mark

Write 8.4×10−48.4\times10^{-4} as an ordinary number.

Show worked answer

A negative index means divide by the corresponding positive power.

10−4=1104=0.000110^{-4}=\frac{1}{10^4}=0.0001 8.4×0.0001=0.000848.4\times0.0001=\boxed{0.00084}

Order standard-form numbers

2 marks

Write these numbers in ascending order:

2.8×105,9.1×104,3.05×105,7.6×1042.8\times10^5,\quad9.1\times10^4,\quad3.05\times10^5,\quad7.6\times10^4
Show worked answer

Numbers with power 10410^4 are smaller than numbers with power 10510^5.

Within each power group, compare the coefficients:

7.6<9.1and2.8<3.057.6<9.1\qquad\text{and}\qquad2.8<3.05

Therefore the ascending order is

7.6×104, 9.1×104, 2.8×105, 3.05×105\boxed{7.6\times10^4,\ 9.1\times10^4,\ 2.8\times10^5,\ 3.05\times10^5}

Multiply and normalise

2 marks

Work out (4.2×106)(3×10−4)(4.2\times10^6)(3\times10^{-4}). Give your answer in standard form.

Show worked answer

Multiply the coefficients and add the indices.

4.2×3=12.64.2\times3=12.6 106×10−4=106+(−4)=10210^6\times10^{-4}=10^{6+(-4)}=10^2 12.6×10212.6\times10^2

This is not standard form because 12.6≥1012.6\ge10. Divide the coefficient by 1010 and increase the index by 11.

12.6×102=1.26×10312.6\times10^2=\boxed{1.26\times10^3}

Estimate: (4×106)(3×10−4)=12×102(4\times10^6)(3\times10^{-4})=12\times10^2, so an answer of order 10310^3 is reasonable.

Divide and normalise

3 marks

Work out 6.4×10−58×102\dfrac{6.4\times10^{-5}}{8\times10^2}. Give your answer in standard form.

Show worked answer

Divide the coefficients and subtract the second index.

6.4÷8=0.86.4\div8=0.8 10−5÷102=10−5−2=10−710^{-5}\div10^2=10^{-5-2}=10^{-7} 0.8×10−70.8\times10^{-7}

The coefficient is below 11. Multiply it by 1010 and decrease the index by 11 to preserve the value.

0.8×10−7=8×10−80.8\times10^{-7}=\boxed{8\times10^{-8}}

The answer must be smaller than 6.4×10−56.4\times10^{-5} because it was divided by 800800.

Add different powers

3 marks

Work out 3.7×106+8.5×1053.7\times10^6+8.5\times10^5. Give your answer in standard form.

Show worked answer

The powers must match before the coefficients can be added.

8.5×105=0.85×1068.5\times10^5=0.85\times10^6

Now both coefficients multiply 10610^6.

3.7×106+0.85×1063.7\times10^6+0.85\times10^6 =(3.7+0.85)×106=(3.7+0.85)\times10^6 4.55×106\boxed{4.55\times10^6}

The answer is just above 3.73.7 million, which fits adding 0.850.85 million.

Subtract different negative powers

3 marks

Work out 5.02×10−3−7.6×10−45.02\times10^{-3}-7.6\times10^{-4}. Give your answer in standard form.

Show worked answer

Match the powers. Be careful: 10−410^{-4} is one tenth of 10−310^{-3}.

7.6×10−4=0.76×10−37.6\times10^{-4}=0.76\times10^{-3} 5.02×10−3−0.76×10−35.02\times10^{-3}-0.76\times10^{-3} =(5.02−0.76)×10−3=(5.02-0.76)\times10^{-3} 4.26×10−3\boxed{4.26\times10^{-3}}

The result is positive and slightly below 5.02×10−35.02\times10^{-3}, so the scale is sensible.

Correct two non-standard forms

2 marks

Write each expression in standard form.

(a) 0.48×1090.48\times10^9 (b) 37×10−637\times10^{-6}

Show worked answer

(a) Multiply the coefficient by 1010, so compensate by reducing the index by 11.

0.48×109=4.8×1080.48\times10^9=\boxed{4.8\times10^8}

(b) Divide the coefficient by 1010, so compensate by increasing the index by 11.

37×10−6=3.7×10−537\times10^{-6}=\boxed{3.7\times10^{-5}}

Both final coefficients now satisfy 1≤a<101\le a<10.

Use a scientific calculator

3 marks

Use a calculator to work out

(2.4×107)(5.5×10−3)3×102\frac{(2.4\times10^7)(5.5\times10^{-3})}{3\times10^2}

Give your answer in standard form.

Show worked answer

Enter each number with the calculator’s ×10x\times10^x or EXP key, keeping the numerator in brackets.

Working with the parts also checks the display:

2.4×5.53=4.4\frac{2.4\times5.5}{3}=4.4 107+(−3)−2=10210^{7+(-3)-2}=10^2 4.4×102\boxed{4.4\times10^2}

A calculator may show 4.4E2; this means 4.4×1024.4\times10^2, or 440440.

Interpret E notation

2 marks

A calculator displays 6.207E−5.

(a) Write this in standard form. (b) Write it as an ordinary number.

Show worked answer

The display E−5 means “multiply by 10−510^{-5}”.

(a)

6.207×10−5\boxed{6.207\times10^{-5}}

(b) Because 10−5=0.0000110^{-5}=0.00001,

6.207×10−5=0.000062076.207\times10^{-5}=\boxed{0.00006207}

The minus sign belongs to the index; it does not make the number negative.

Data transfer context

3 marks

A research station sends 3.2×1063.2\times10^6 bytes of data each second for 2.5×1022.5\times10^2 seconds. Calculate the total number of bytes sent. Give your answer in standard form.

Show worked answer

Total data equals rate multiplied by time. The seconds cancel, leaving bytes.

data=(3.2×106)(2.5×102)\text{data}=(3.2\times10^6)(2.5\times10^2) =(3.2×2.5)×106+2=(3.2\times2.5)\times10^{6+2} =8×108=8\times10^8 8×108 bytes\boxed{8\times10^8\text{ bytes}}

Estimate: 3×106×3×102=9×1083\times10^6\times3\times10^2=9\times10^8, so the order of magnitude is reasonable.

Microscopic mass context

3 marks

One microscopic organism has a mass of 7.5×10−87.5\times10^{-8} g. A sample contains 4×1054\times10^5 organisms. Find the total mass in grams, in standard form.

Show worked answer

Multiply the mass of one organism by the number of organisms.

total mass=(7.5×10−8)(4×105)\text{total mass}=(7.5\times10^{-8})(4\times10^5) =30×10−3=30\times10^{-3}

Normalise the coefficient:

30×10−3=3×10−230\times10^{-3}=3\times10^{-2} 3×10−2 g\boxed{3\times10^{-2}\text{ g}}

This is 0.030.03 g, so it is sensible that many microscopic organisms still have a small total mass.

Convert a very small measurement

3 marks

A dust particle has a diameter of 6.3×10−76.3\times10^{-7} m. Convert this diameter to millimetres. Give your answer in standard form.

Show worked answer

There are 10310^3 millimetres in one metre, so multiply by 10310^3.

6.3×10−7 m×103mmm6.3\times10^{-7}\text{ m}\times10^3\frac{\text{mm}}{\text{m}} =6.3×10−7+3 mm=6.3\times10^{-7+3}\text{ mm} 6.3×10−4 mm\boxed{6.3\times10^{-4}\text{ mm}}

Changing from metres to the smaller millimetre unit makes the numerical value larger: the index rises from −7-7 to −4-4.

Examiner-style feedback

Common standard form mistakes

Coefficient outside the allowed range

42×10542\times10^5 and 0.42×1070.42\times10^7 have the right shape but are not standard form. Normalise until 1≤a<101\le a<10.

Wrong sign on the index

A positive number between 0 and 1 needs a negative index. Remember 10−4=1÷10410^{-4}=1\div10^4; the negative sign describes division, not a negative value.

Decimal movement in the wrong direction

Reconnect the shortcut to the calculation: a positive power multiplies the coefficient and makes it larger; a negative power divides it and makes it smaller.

Adding coefficients with different powers

3×106+4×1053\times10^6+4\times10^5 is not 7×10117\times10^{11} or 7×1067\times10^6. First rewrite 4×105=0.4×1064\times10^5=0.4\times10^6.

Using multiplication laws for addition

Indices add only when powers are multiplied: 10a×10b=10a+b10^a\times10^b=10^{a+b}. There is no matching rule for 10a+10b10^a+10^b.

Misreading calculator notation

`2.9E−6` means 2.9×10−62.9\times10^{-6}. Treat the signed number after E as the index, not as a separate operation.

Losing units or using the wrong scale

Carry units through the calculation and write them in the final answer. For a conversion, decide whether the numerical value should become larger or smaller before changing the power.

30-SECOND RECAP

Coefficient, power, operation, check

Standard form separates significant digits from scale. Let the operation determine what happens to the coefficients and indices, then finish by checking the form and the size.

  1. For a positive number, keep the coefficient at least 1 but less than 10.
  2. Use a positive index for large values and a negative index for values between 0 and 1.
  3. Multiply/divide: use index laws. Add/subtract: match powers first.
  4. Normalise, keep units and estimate the order of magnitude.
Real search questions

Standard form FAQ

What does standard form mean in maths?

Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. It is a compact way to show very large and very small numbers.

How do you write a number in standard form?

First create a coefficient between 1 and 10 from the significant digits. Then choose the power of 10 that restores the original place value. Check by multiplying the coefficient by that power.

Why do small numbers have a negative power?

A negative index represents a reciprocal. For example, 10⁻³ = 1/10³ = 0.001, so it reduces a coefficient between 1 and 10 to a number between 0 and 1.

How do you multiply numbers in standard form?

Multiply the coefficients, add the indices of the powers of 10, then adjust the result if the coefficient is not between 1 and 10.

How do you add standard-form numbers with different powers?

Rewrite one number so both terms use the same power of 10. Only then add the coefficients, because they must represent the same place-value unit.

What does E mean on a calculator?

E is calculator shorthand for × 10 to a power. For example, 4.7E−6 means 4.7 × 10⁻⁶. It does not mean multiply by the mathematical constant e.

Is standard form the same as scientific notation?

In UK school maths, standard form is the usual name. In many other countries the same normalised a × 10ⁿ notation is called scientific notation.

Build the skill

Prerequisites and useful next topics

Revise first

Powers, indices and place value

Be secure with integer powers of 10, negative indices, decimal place value and multiplying or dividing by powers of 10.

Use it next

Estimation, units and formulae

Standard form combines naturally with significant figures, estimation, unit conversion and substitution into scientific formulae.

Browse all GCSE Maths study guidesPractise calculations in simultaneous equationsUse calculator skills in trigonometry
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This course is for Higher-tier learners. Foundation learners can continue with the original questions and GCSE Maths guides.

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Content standards

Curriculum and rights review

Curriculum references checked 4 September 2026. Standard form is included for both Foundation and Higher in the specifications reviewed. All questions, values, contexts and solution wording on this page are original Pass an Exam content.