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Maths

Index Laws: Multiplying and Dividing Powers

Overview

Students discover the index laws through pattern-spotting with expanded forms before applying them to algebraic expressions.

Learning Objective
Students understand and apply the multiplication and division laws of indices, including the zero index, and simplify expressions involving powers.

Resources needed

  • Mini whiteboards
  • Pattern investigation sheet (or board)

Lesson stages

0 / 7 done
  1. 1 Write 2³ × 2⁴. Expand: (2×2×2) × (2×2×2×2) = 2⁷. 'How many 2s do we have? 3+4=7. So 2³ × 2⁴ = 2⁷.' Try 3² × 3⁵. Students expand and count. What's the pattern?
  2. 2 Generalise: aᵐ × aⁿ = aᵐ⁺ⁿ. 'When multiplying powers with the same base, add the indices.' Students practise: 5³ × 5⁶, x² × x⁵, y⁴ × y.
  3. 3 Write 2⁵ ÷ 2². Expand: (2×2×2×2×2)/(2×2). Cancel: 2×2×2 = 2³. 'We subtract: 5−2=3.' Generalise: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Practice: 3⁷ ÷ 3⁴, x⁶ ÷ x², p⁵ ÷ p³.
  4. 4 Use division law: 3³ ÷ 3³ = 3⁰. But also: any number ÷ itself = 1. So 3⁰ = 1. 'Any non-zero number to the power zero equals 1.' Check with students: 100⁰ = ?, x⁰ = ?, 0.5⁰ = ?
  5. 5 Simplify: 3x² × 4x³ = 12x⁵ (multiply coefficients, add indices). Simplify: 6y⁷ ÷ 2y⁴ = 3y³ (divide coefficients, subtract indices). Students practise 6 mixed examples.
  6. 6 Display: a² × b² = ab⁴ (False — different bases). / x³ × x⁰ = x³ (True). / 2⁴ ÷ 2 = 2³ (True — 2=2¹). Students classify and correct the false ones.
  7. 7 If x³ × xⁿ = x⁷, find n. If 2ⁿ ÷ 2³ = 2⁴, find n. If 5ⁿ = 1, find n. Algebra within indices.

Tap a step to mark it as done.

Variations

  • Extend to the power of a power law: (aᵐ)ⁿ = aᵐⁿ
  • Include negative indices (a⁻ⁿ = 1/aⁿ)
  • Use in scientific notation context
More information

Display terms: base, index (pl. indices), power, coefficient. Write out the laws in words as well as symbols: 'when multiplying, add the powers.'

Focus on numerical bases (2, 3, 5) before introducing algebraic bases. Provide an index law reference card. Always expand first to verify.

Do students add (not multiply) indices when multiplying? Do they apply laws only when bases are the same? Do they correctly handle the zero index?

All practice on mini whiteboards. Expand and verify numerically if unsure — this is a reliable self-checking strategy.

Students may apply multiplication law across different bases (x² × y³ = xy⁵). Emphasise repeatedly: the law only works when the bases are identical.

Activity details
Secondary Medium 40 minutes 5 minutes prep Solo, Pairs Indoor
Topics
Indices Powers Algebra Laws Of Indices