lesson
Keep an exact form until approximation is requested
1 · Learn
An exact answer preserves the stated value: a simplified fraction, a multiple of π or a surd. A decimal produced by a calculator is usually an approximation. Do not replace 5π by 15.7 unless a degree of accuracy is requested.
Standard form writes a non-zero number as A×10^n with 1≤|A|<10 and integer n. When multiplying, multiply the A values and add exponents, then renormalise if the new A is outside the required interval.
With fractions, use a common denominator for addition and subtraction. Exact calculation is not the same as refusing to simplify: 3/4+5/6=9/12+10/12=19/12.
2 · Worked example
(3×10^5)(4×10^-2)=12×10^3=1.2×10^4. A circle of radius 3 has circumference 6π exactly; to 3 significant figures it is 18.8.
3 · Your turn
Calculate 5/8+7/12 exactly. Then write (6×10^4)(3×10^-7) in standard form.
Check your answer
29/24; 1.8×10^-2.
4 · Apply your learning
Sort supplied answers into exact and approximate forms. For each approximation, write the rounding instruction needed to justify it.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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