Mathematics • Year 10 • Unit 1 • Lesson 8
Index Laws (Multiplication and Division) — Mixed Challenge
Pull together both laws from Lesson 8: aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Choose the right tool for each problem, spot a classic mistake, and tackle an open-ended challenge.
1. Mixed problems — choose the right law
Each question uses a different combination of Index Laws 1 and 2. Decide which law applies before writing. Show working. 3 marks each
1.1 Simplify (6a⁷) × (2a³) ÷ (4a⁴). Leave in index form.
1.2 Simplify (15x⁹y⁴) ÷ (3x²y). State any restrictions on x and y.
1.3 Find n if 7¹² ÷ 7ⁿ = 7⁵. Check your answer by also computing 7ⁿ as an indexed value.
1.4 Simplify (m⁵ × m⁸) ÷ (m³ × m²). Show every step.
1.5 A claim. "3⁴ × 2⁴ = 6⁸." Show this is wrong with a quick numerical check, and state which condition for Index Law 1 fails.
1.6 Simplify ((8p³q⁵) × (3p²q²)) ÷ (6p⁴q⁶). Leave in index form.
2. Find the mistake
A student has tried to simplify (4x⁶) × (3x²) ÷ (2x³). Their working is below. Exactly one line contains a mistake. Spot it, explain why, then re-do correctly. 3 marks
Student's working — simplify (4x⁶) × (3x²) ÷ (2x³):
Line 1: (4x⁶) × (3x²) = 12 × x⁶ × x² = 12x⁸
Line 2: (12x⁸) ÷ (2x³) = (12 ÷ 2) × (x⁸ ÷ x³)
Line 3: = 6 × x^(8 ÷ 3)
Line 4: = 6 × x^2.67
Line 5: ≈ 6x^2.67
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? The lesson's Index Law 2 says aᵐ ÷ aⁿ = aᵐ⁻ⁿ — that's a minus sign, not a divide sign.3. Open-ended challenge — same answer, two ways
This question has more than one valid answer. 4 marks
3.1 Find two different expressions, each in the form aᵐ × aⁿ (with a, m, n all positive whole numbers and a > 1), that simplify to 2¹⁰. Then find one expression in the form aᵐ ÷ aⁿ that also simplifies to 2¹⁰.
For each expression you give:
(i) Write it down.
(ii) Show the working that confirms it equals 2¹⁰.
(iii) State whether you used Index Law 1 or 2.
Constraint: Your three expressions must be different from each other and must NOT include the trivial 2¹⁰ × 2⁰ or 2¹⁰ ÷ 2⁰.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — (6a⁷) × (2a³) ÷ (4a⁴)
Numerator: 6a⁷ × 2a³ = 12 × a⁷⁺³ = 12a¹⁰.
Divide: 12a¹⁰ ÷ 4a⁴ = 3 × a¹⁰⁻⁴ = 3a⁶.
1.2 — (15x⁹y⁴) ÷ (3x²y)
(15 ÷ 3) × (x⁹ ÷ x²) × (y⁴ ÷ y¹) = 5 × x⁹⁻² × y⁴⁻¹ = 5x⁷y³, with x ≠ 0 and y ≠ 0 (else the original is undefined).
1.3 — 7¹² ÷ 7ⁿ = 7⁵
By Law 2, 7¹² ÷ 7ⁿ = 7¹²⁻ⁿ. Match: 12 − n = 5, so n = 7.
Check: 7¹² ÷ 7⁷ = 7¹²⁻⁷ = 7⁵ ✓.
1.4 — (m⁵ × m⁸) ÷ (m³ × m²)
Numerator: m⁵ × m⁸ = m⁵⁺⁸ = m¹³.
Denominator: m³ × m² = m³⁺² = m⁵.
Divide: m¹³ ÷ m⁵ = m¹³⁻⁵ = m⁸.
1.5 — Claim: 3⁴ × 2⁴ = 6⁸
Numerical check: 3⁴ = 81; 2⁴ = 16; 81 × 16 = 1,296. But 6⁸ = 1,679,616 — wildly different.
Condition that fails: the bases must be the same to use Index Law 1. Here the bases are 3 and 2, which differ, so the law does not apply. (Note: 3⁴ × 2⁴ = (3 × 2)⁴ = 6⁴ = 1,296 is true — that's a different index law from Lesson 9, the power-of-a-product rule.)
1.6 — ((8p³q⁵) × (3p²q²)) ÷ (6p⁴q⁶)
Numerator: 8p³q⁵ × 3p²q² = (8 × 3) × p³⁺² × q⁵⁺² = 24p⁵q⁷.
Divide: 24p⁵q⁷ ÷ 6p⁴q⁶ = (24 ÷ 6) × p⁵⁻⁴ × q⁷⁻⁶ = 4pq (i.e. 4p¹q¹).
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The student divided the indices instead of subtracting them. Index Law 2 says aᵐ ÷ aⁿ = aᵐ⁻ⁿ — subtract, not divide. So x⁸ ÷ x³ should be x⁸⁻³ = x⁵, not x^(8÷3).
(c) Corrected working:
(4x⁶) × (3x²) = 12x⁸
(12x⁸) ÷ (2x³) = (12 ÷ 2) × (x⁸ ÷ x³)
= 6 × x⁸⁻³
= 6x⁵.
This is one of the most common index-law slips — "÷ of indices" gets confused with "subtract indices".
3 — Open-ended challenge (sample solution)
We need aᵐ × aⁿ = 2¹⁰, so the bases must equal 2 and we need m + n = 10.
Expression 1 (Law 1): 2³ × 2⁷.
Working: 2³ × 2⁷ = 2³⁺⁷ = 2¹⁰ ✓. Law used: Index Law 1.
Expression 2 (Law 1): 2⁴ × 2⁶.
Working: 2⁴ × 2⁶ = 2⁴⁺⁶ = 2¹⁰ ✓. Law used: Index Law 1.
Expression 3 (Law 2): 2¹⁵ ÷ 2⁵.
Working: 2¹⁵ ÷ 2⁵ = 2¹⁵⁻⁵ = 2¹⁰ ✓. Law used: Index Law 2.
Other valid combinations: 2² × 2⁸, 2¹ × 2⁹, 2⁵ × 2⁵ for Law 1; 2¹² ÷ 2², 2²⁰ ÷ 2¹⁰ for Law 2.
Marking: 1 mark per valid Law-1 expression (up to 2); 2 marks for the valid Law-2 expression with correct working. Total 4. Award full marks for any three distinct valid expressions.