This checkpoint assesses your ability to apply the index laws to algebraic expressions: writing and reading algebraic powers, the product rule with coefficients and variables, the quotient and power rules, and mixed simplification. It covers Lessons 11–14.
❓ Multiple Choice (10 questions)
1. Simplify x³ × x⁴ using the product rule.
2. In the term 5a², what is the coefficient?
3. Simplify 2a³ × 3a².
4. Simplify 12x⁵ ÷ 4x² using the quotient rule.
5. Simplify (2x³)².
6. Simplify x⁶ × x (remember x means x¹).
7. Simplify 15a⁴b³ ÷ 3a²b.
8. Simplify (3y²)³.
9. Simplify (2x²)³ × x⁴ ÷ 4x³.
10. Simplify 6a⁵b² ÷ (2a²b² × 3a).
✍ Short Answer (4 questions)
11. Simplify each expression, leaving your answer in index form.
(a) 4x³ × 2x⁵ (2 marks)
(b) 20a⁶ ÷ 5a² (2 marks)
4 MARKS
12. Use the power-of-a-power rule to expand each expression.
(a) (3x⁴)² (2 marks)
(b) (2a²b)³ (2 marks)
(c) Explain why the coefficient is raised to the power as well as each variable. (1 mark)
5 MARKS
13. Simplify each of the following, using the index laws.
(a) 12x⁵y³ ÷ 4x²y (2 marks)
(b) (2x³)² × x (1 mark)
(c) 6a² × 3a⁴ ÷ 2a³ (1 mark)
(d) Explain why 5x³ × 4y² cannot be combined into a single power. (1 mark)
5 MARKS
14. Mixed simplification: Simplify the expression (4x²y³)² × 2x³ ÷ (8x⁵y⁴), step by step.
(a) Expand (4x²y³)² using the power-of-a-power rule. (2 marks)
(b) Multiply your result by 2x³ using the product rule. (1 mark)
(c) Divide the coefficients (the numbers) when dividing by 8x⁵y⁴. (1 mark)
(d) Apply the quotient rule to the powers of x. (1 mark)
(e) Apply the quotient rule to the powers of y. (1 mark)
(f) Write the fully simplified expression. (1 mark)
(g) Name the index law used at each of stages (a), (b) and (c)–(e), and state why the bases must match for the product and quotient rules. (2 marks)
9 MARKS
1. DProduct rule: same base, add indices. x³ × x⁴ = x³⁺⁴ = x⁷ (the base x stays; do not multiply the indices or double the base).
2. AThe coefficient is the number multiplying the variable, so in 5a² it is 5 (2 is the index).
3. CMultiply coefficients and add the indices: (2 × 3)a³⁺² = 6a⁵.
4. BDivide coefficients and subtract indices: (12 ÷ 4)x⁵⁻² = 3x³.
5. APower of a power: raise the coefficient and multiply each index. (2x³)² = 2²x³ˣ² = 4x⁶.
6. Cx = x¹, so x⁶ × x¹ = x⁶⁺¹ = x⁷.
7. BDivide coefficients (15 ÷ 3 = 5), then subtract indices for each variable: a⁴⁻² b³⁻¹ = 5a²b².
8. DCube the coefficient and multiply each index by 3: (3y²)³ = 3³y²ˣ³ = 27y⁶.
9. C(2x²)³ = 8x⁶; × x⁴ = 8x¹⁰; ÷ 4x³ = (8 ÷ 4)x¹⁰⁻³ = 2x⁷.
10. ADenominator: 2a²b² × 3a = 6a³b². Then 6a⁵b² ÷ 6a³b² = a⁵⁻³ b²⁻² = a² × b⁰ = a².
Q11 (4 marks): (a) Multiply coefficients (4 × 2 = 8) and add indices: 4x³ × 2x⁵ = 8x⁸ [2]. (b) Divide coefficients (20 ÷ 5 = 4) and subtract indices: 20a⁶ ÷ 5a² = 4a⁴ [2].
Q12 (5 marks): (a) (3x⁴)² = 3²x⁴ˣ² = 9x⁸ [2]. (b) (2a²b)³ = 2³a²ˣ³b¹ˣ³ = 8a⁶b³ [2]. (c) The whole bracket is being multiplied by itself that many times, so every factor inside, the coefficient and each variable, is raised to the power [1].
Q13 (5 marks): (a) 12x⁵y³ ÷ 4x²y = (12 ÷ 4)x⁵⁻²y³⁻¹ = 3x³y² [2]. (b) (2x³)² × x = 4x⁶ × x = 4x⁷ [1]. (c) 6a² × 3a⁴ ÷ 2a³ = 18a⁶ ÷ 2a³ = 9a³ [1]. (d) The product rule only adds indices when the bases are the same; 5x³ and 4y² have different variables (x and y), so they stay as a single product 20x³y² and cannot become one power [1].
Q14 (9 marks): (a) (4x²y³)² = 4²x²ˣ²y³ˣ² = 16x⁴y⁶ [2]. (b) 16x⁴y⁶ × 2x³ = (16 × 2)x⁴⁺³y⁶ = 32x⁷y⁶ [1]. (c) 32 ÷ 8 = 4 [1]. (d) x⁷ ÷ x⁵ = x⁷⁻⁵ = x² [1]. (e) y⁶ ÷ y⁴ = y⁶⁻⁴ = y² [1]. (f) Fully simplified: 4x²y² [1]. (g) Stage (a) = power-of-a-power rule (multiply indices); stage (b) = product rule (add indices); stages (c)–(e) = quotient rule (divide coefficients, subtract indices). The product and quotient rules only combine indices when the bases match, which is why x and y are handled separately [2].
Tick when you have finished all questions and checked your answers.