Calculations in Standard Form (AI SL)
Once numbers are written in standard form, the GDC-friendly rules for combining them are different from ordinary arithmetic - you work with the coefficient and the power of 10 separately. This page covers multiplying, dividing, adding and subtracting numbers of the form \(a\times10^k\), with worked examples and the mistakes that cost the most marks. It's part of the broader Standard Form topic.
57 questions on this sub-topic.
Multiplying, dividing, adding and subtracting
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. None of these rules are in the formula booklet - they follow directly from the index laws you already know.
Multiplying
\((a\times10^m)\times(b\times10^n)=ab\times10^{m+n}\)
Multiply the coefficients, add the powers of 10.
Dividing
\(\dfrac{a\times10^m}{b\times10^n}=\dfrac{a}{b}\times10^{m-n}\)
Divide the coefficients, subtract the powers of 10.
Adding or subtracting
Rewrite so both powers of 10 match, then add or subtract the coefficients
You can't combine coefficients until the powers line up - there's no shortcut around this step.
Need the full syllabus wording and formula-booklet reference table? See Standard Form.
Worked examples
Evaluate \((5.2\times10^{6}) + (8\times10^{5})\), in standard form.
Worked solution
You can only add the coefficients once the powers of ten are equal. Rewrite the smaller-power term: \(8\times10^{5} = 0.8\times10^{6}.\) M1
\(5.2 + 0.8 = 6.0.\) A1
\(6.0\times10^{6}.\) The coefficient already satisfies \(1\le a<10\), so it is in standard form. A1
Evaluate \(\dfrac{(8\times10^{3})(3\times10^{-5})}{6\times10^{-4}}\), in standard form.
Worked solution
Multiply coefficients and add indices: \((8\times10^{3})(3\times10^{-5}) = 24\times10^{3+(-5)}.\) M1
\(24\times10^{3+(-5)} = 24\times10^{-2}.\) A1
\(6\times10^{-4}\):
Coefficients \(\dfrac{24}{6}=4\); indices \(10^{-2-(-4)} = 10^{2}.\) \(4\times10^{2}.\) The coefficient is in range, so this is the final standard form. A1
Given \(p = 6.0 \times 10^{8}\) and \(q = 1.5 \times 10^{-3}\), evaluate, giving each answer in standard form:
(a) \(pq\)
(b) \(\dfrac{p}{q}\)
Worked solution
(a) \(pq = (6.0\times1.5)\times10^{8+(-3)}.\) M1
\(pq = 9.0\times10^{5}.\) A1
(b) \(\dfrac{p}{q} = \dfrac{6.0}{1.5}\times10^{8-(-3)}.\) M1
\(\dfrac{p}{q} = 4.0\times10^{11}.\) A1
Light travels at \(3.00 \times 10^{8}\) m s\(^{-1}\). The distance from the Sun to Earth is \(1.50 \times 10^{11}\) m.
(a) Find the time, in seconds, for light to travel this distance.
(b) Find this time in minutes.
Worked solution
(a) Time \(= \dfrac{1.50\times10^{11}}{3.00\times10^{8}}.\) M1
Time \(= 500\) s. A1
(b) \(500 \div 60\) converts seconds to minutes. M1
\(500 \div 60 \approx 8.33\) minutes. A1
Common mistakes
- Leaving the coefficient outside \(1\le a<10\). A multiplication or power result like \(25.2\times10^{4}\) is correct arithmetic but not standard form - it must be renormalised to \(2.52\times10^{5}\).
- Adding or subtracting before matching the powers of 10. \(5.2\times10^{6} + 8\times10^{5}\) is not \(13.2\times10^{11}\) - rewrite one term so the powers agree first, then combine the coefficients.
- Mixing up the index rule for division. Dividing standard form numbers subtracts the powers of 10 (\(10^{m-n}\)), not adds them - the same slip that appears when dividing ordinary indices.
Ready to practise properly?
59 standard-form calculation questions, marked instantly like the real exam.
Quick answers
How do you multiply or divide numbers in standard form?
Multiply or divide the coefficients as normal, then add (for multiplication) or subtract (for division) the powers of 10. Renormalise afterwards if the coefficient falls outside \(1\le a<10\).
Can you add two numbers in standard form directly?
Only if the powers of 10 already match. If they don't, rewrite one term so both share the same power before adding or subtracting the coefficients.