Maths Lab · Free lesson · 22 minutes
Indices describe structure, not shortcuts
Derive index laws from repeated factors and interpret zero, negative and fractional powers with their domain restrictions.
SS1 · indices01 · Think first
Start with a question.
Walking down the powers of 2 gives 8, 4, 2, 1, then what? A negative exponent continues the division pattern; it does not change the sign of the base.
What you already know
- Repeated multiplication
- Divide fractions
- Square and cube roots of positive numbers
02 · See it
The power ladder for base 2
Each step down divides by 2. The ladder passes through 1 and continues into reciprocals.
- 2³
- 8
- 2²
- 4
- 2¹
- 2
- 2⁰
- 1
- 2⁻¹
- 1/2
- 2⁻²
- 1/4
Use the justified worked steps below to explore this idea.
03 · Show steps · explain why
One justified step at a time.
- 1
a³ × a² = (a × a × a)(a × a) = a⁵
For the same base, multiplication joins factor strings. This is why the exponents add.
04 · Mistake clinic
Find the first wrong idea.
2⁻³ = −8; x² + x³ = x⁵.
Compare the correction and its reason
2⁻³ = 1/8. In general, x² + x³ stays x²(1 + x).
The reciprocal rule concerns a negative exponent, not a negative value. Adding exponents is a multiplication law, not an addition law.
Check: Set x = 2: x² + x³ = 4 + 8 = 12, while x⁵ = 32. One counterexample disproves the claimed identity.
A mistake is evidence about the next idea to repair. It is not a label for the learner.
05 · Your turn
Try first. Then use a hint.
06 · Check & reflect → next step
Understanding includes explaining.
Derive one law using factors. Name one case where that law cannot be applied.
Use powers to understand repeated percentage growth. Keep bases, exponents and units distinct.
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JEM-S1-M01-002 · Phase 4A SS1 General Mathematics · Lesson 4 · printed pp. 20–21
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