lesson
Count factors before using the shortcut
1 · Learn
For powers with the same non-zero base, multiplying adds exponents and dividing subtracts them. This follows from counting repeated factors and cancelling equal non-zero factors. Raising a power to a power multiplies the exponents.
These rules do not mean that a² + a³ equals a⁵: addition is a different operation. Keep the base and the operation visible.
2 · Worked example
2³ × 2² = (2×2×2) × (2×2) = 2⁵ = 32. By contrast, 2³ + 2² = 8 + 4 = 12.
3 · Your turn
Simplify x⁴ × x³ and x⁶ / x² for x not equal to zero.
Check your answer
x⁷ and x⁴.
4 · Apply your learning
Expand one example into factors to justify each rule and supply a counterexample to the incorrect addition rule.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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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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Year 8 Maths learning routine
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