Calculations in Standard Form (AA SL)
Once numbers are written as \(a\times10^{k}\), you can multiply, divide, add, subtract and raise them to powers without ever expanding back to their ordinary decimal form - provided you keep the coefficient and the power of ten separate, and re-normalise at the end. This page covers the index-law shortcuts that make this fast, and the answer must always be given back in standard form. It's part of the broader Standard Form topic.
28 questions on this sub-topic.
Key rules
Covered under IB syllabus reference SL1.1: operations with numbers in the form \(a\times10^{k}\) where \(1\le a<10\) and \(k\) is an integer. A final answer left in calculator or computer notation, such as \(5.2\text{E}30\) instead of \(5.2\times10^{30}\), is not acceptable.
Multiply and divide
\(10^{a}\times10^{b} = 10^{a+b}\), \(\quad 10^{a}\div10^{b} = 10^{a-b}\)
Multiply or divide the coefficients as ordinary numbers, then add (multiplying) or subtract (dividing) the powers of ten.
Powers of a standard-form number
\((a\times10^{m})^{n} = a^{n}\times10^{mn}\)
Not in the formula booklet - it follows from the ordinary power-of-a-power rule applied to each factor separately.
Need the full syllabus wording and formula-booklet reference table? See Standard Form.
Worked examples
Evaluate \((3\times10^{4})(2\times10^{6})\), giving your answer in standard form.
Worked solution
\(3\times 2 = 6.\) M1
(\(10^a\times 10^b = 10^{a+b}\)):
\(10^{4+6} = 10^{10}\Rightarrow6\times10^{10}.\) A1
Find \((4.2 \times 10^{6}) + (8 \times 10^{5})\), giving your answer in standard form.
Worked solution
To add, both terms need the same power; rewrite \(8\times 10^{5} = 0.8\times 10^{6}.\) M1
\(4.2 + 0.8 = 5.0\times10^{6}.\) A1
One molecule has mass \(3.0\times 10^{-23}\) g.
(a) Find the mass of \(6.0\times 10^{23}\) molecules.
(b) Find how many molecules are in \(45\) g, in standard form to 3 significant figures.
Worked solution
(a) \((3.0\times 10^{-23})(6.0\times 10^{23}) = (3.0\times 6.0)\times 10^{-23+23}.\) M1
\(= 18\times 10^{0} = 18\) g. A1
(b) \(\dfrac{45}{3.0\times 10^{-23}}.\) M1
\(= 15\times 10^{23} = 1.5\times 10^{24}\) molecules. A1
Common mistakes
- Adding powers of ten when adding or subtracting numbers. The \(10^{a+b}\) index law only applies to multiplication and division - for \(+\) and \(-\), the powers of ten must first be made equal, then only the coefficients are combined.
- Forgetting to re-normalise the answer. A computation such as \(24\times10^{3}\) is correct arithmetic but not standard form, since \(24\ge10\) - it must be rewritten as \(2.4\times10^{4}\) before it is accepted.
- Leaving the coefficient outside \(1\le a<10\). \(45\times10^{3}\) is correct arithmetic but not standard form - it must be rewritten as \(4.5\times10^{4}\).
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Quick answers
How do you multiply or divide numbers in standard form?
Multiply or divide the coefficients as normal, then add the powers of ten when multiplying or subtract them when dividing, using \(10^{a}\times10^{b}=10^{a+b}\). Check the coefficient is still between 1 and 10 and adjust if it is not.
How do you add or subtract numbers in standard form?
First rewrite both numbers with the same power of ten, then add or subtract the coefficients only, leaving the shared power of ten unchanged. Re-convert to standard form afterwards if the coefficient falls outside 1 to 10.