lesson
A power counts repeated factors
1 · Learn
A power such as 3³ means 3 × 3 × 3, not 3 × 3. A square root reverses squaring for a non-negative value; the principal square root of 49 is 7. Prime factorisation expresses a whole number as a product of primes.
Factor trees may branch differently but should reach the same prime factors. Use those factors to identify common factors or multiples, keeping their repetitions.
2 · Worked example
For 12 = 2² × 3 and 18 = 2 × 3², the highest common factor is 2 × 3 = 6 and the lowest common multiple is 2² × 3² = 36.
3 · Your turn
Write 72 as a product of primes and evaluate 2⁴.
Check your answer
72 = 2³ × 3²; 2⁴ = 16.
4 · Apply your learning
Make two different factor trees for 72 and compare the final prime factors.
Adult guidance and safety
Require a reasoned method and an independent check. Use accurate number lines, geometric drawings and data displays where the task needs them. These text models require varied practice and diagnostic assessment before curriculum approval.
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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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