How Do I Calculate with Numbers in Standard Form?
Learn how to multiply, divide, add and subtract numbers in standard form, with clear GCSE methods, worked examples and common mistakes to avoid.
📋 Spec coverage: GCSE Edexcel Mathematics (1MA1) This article covers Edexcel N9 - Standard form for Foundation and Higher tiers: calculating with and interpreting standard form. It may be assessed on Paper 1, Paper 2 or Paper 3.
To calculate with numbers in standard form, calculate using the first numbers and apply the appropriate index law to the powers of ten. Your final answer must have the form
A×10nA \times 10^nA×10nwhere 1≤A<101 \leq A < 101≤A<10 and nnn is an integer.
Standard form makes very large and very small numbers easier to compare and calculate with. The number AAA is called the coefficient, and nnn is the power, or index, of ten.
Multiplication and division
For multiplication, multiply the first numbers and add the powers:
(3×104)(2×103)=6×104+3=6×107(3 \times 10^4)(2 \times 10^3) = 6 \times 10^{4+3} = 6 \times 10^7(3×104)(2×103)=6×104+3=6×107Sometimes a calculation produces an answer that is not yet in standard form:
(4×102)(3×105)=12×107=1.2×108(4 \times 10^2)(3 \times 10^5) =12 \times 10^7 =1.2 \times 10^8(4×102)(3×105)=12×107=1.2×108Because 121212 is not less than 101010, move the decimal point one place left and increase the power by 111.
For division, divide the first numbers and subtract the powers:
8×1072×103=4×107−3=4×104\frac{8 \times 10^7}{2 \times 10^3} = 4 \times 10^{7-3} = 4 \times 10^42×1038×107=4×107−3=4×104Addition and subtraction
Before adding or subtracting, rewrite the numbers so they have the same power of ten.
For example:
3.2×105+4×1043.2 \times 10^5 + 4 \times 10^43.2×105+4×104Rewrite 4×1044 \times 10^44×104 as 0.4×1050.4 \times 10^50.4×105:
3.2×105+0.4×105=3.6×1053.2 \times 10^5 + 0.4 \times 10^5 = 3.6 \times 10^53.2×105+0.4×105=3.6×105The same-power rule also works for subtraction.
Check the final answer
If the first number is not at least 111 but less than 101010, adjust it. For example:
24×106=2.4×10724 \times 10^6 = 2.4 \times 10^724×106=2.4×107A positive power represents a large number, while a negative power represents a number between 000 and 111. For example, 5×10−3=0.0055 \times 10^{-3}=0.0055×10−3=0.005.
Estimate first: the result should have a sensible size. This quick size check can catch incorrect indices or misplaced decimal points.
Common exam mistake: Do not add the powers when adding two numbers. Powers are added only when multiplying powers of ten. Also check that the final coefficient satisfies 1≤A<101 \leq A < 101≤A<10.
Completeness check: This covers interpreting standard form, calculating using all four operations, applying index laws and rewriting answers in correct standard form.
