Indices (also called exponents or powers) are a compact way of writing repeated multiplication. When you see 2³, that means 2 × 2 × 2 = 8. Understanding how indices work is the foundation for logarithms, surds, and algebra at WAEC level.
The basic laws
There are six laws you need to know cold: • Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add the powers when the base is the same) • Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the powers) • Power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply the powers) • Zero index: a⁰ = 1 for any non-zero value of a • Negative index: a⁻ⁿ = 1/aⁿ (flip to the denominator) • Fractional index: a^(1/n) = ⁿ√a (the denominator becomes the root)
Worked example 1
Simplify: (2³ × 2⁵) ÷ 2⁴ Step 1: Apply the multiplication law first: 2³ × 2⁵ = 2⁸ Step 2: Apply the division law: 2⁸ ÷ 2⁴ = 2⁴ Step 3: Evaluate: 2⁴ = 16 Answer: 16
Worked example 2
Simplify: (27)^(2/3) Step 1: Rewrite using the fractional index rule: (27)^(2/3) = (³√27)² Step 2: Find the cube root: ³√27 = 3 Step 3: Square the result: 3² = 9 Answer: 9
Common mistakes to avoid
• aᵐ × bⁿ ≠ (ab)ᵐ⁺ⁿ: you can only add powers when the BASE is the same. • 2³ + 2⁵ ≠ 2⁸: the multiplication law applies to ×, not +. • a⁰ = 1, NOT 0. Students frequently write 5⁰ = 0. It equals 1.
Key points
- Add powers when multiplying same base
- Subtract powers when dividing same base
- Any number to the power 0 equals 1
- Negative power means reciprocal (flip it)
- Fractional power (1/n) means the nth root
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WAEC tip
WAEC often tests fractional and negative indices together in one question. Practice writing a⁻²/³ in surd form before the exam: it comes up more than you'd expect.
Practise this
Simplify: (16)^(3/4) × 2⁻³. Write out each step using the laws above.
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