This quiz assesses your understanding of the entire unit: index notation, the index laws, the zero and negative indices, algebraic index laws, scientific notation and its operations, and fractional indices. Show all working for the short answer and extended response questions.
Multiple Choice
1. In the expression 6⁵, what is the index (power)?
2. Evaluate 3³.
3. Simplify 4² × 4⁵ using the product rule.
4. Simplify 9⁶ ÷ 9² using the quotient rule.
5. Simplify (5³)² using the power-of-a-power rule.
6. Simplify 2⁶ × 2² ÷ 2³, leaving your answer as a single power of 2.
7. Evaluate 8⁰.
8. Write 4⁻¹ as a fraction.
9. Evaluate 2⁻³ as a fraction.
10. Simplify 3x² × 4x³.
11. Simplify 20a⁷ ÷ 5a³.
12. Simplify (2y³)².
13. Write 0.00056 in scientific notation.
14. Calculate (3 × 10⁵) × (2 × 10⁴), giving the answer in scientific notation.
15. Evaluate the fractional index 25^(1/2).
Short Answer
16. Use the numerical index laws.
(a) Simplify 5⁴ × 5² ÷ 5³, leaving your answer as a single power of 5. (2 marks)
(b) Evaluate 2⁻² + 3⁰, giving your answer as a fraction. (2 marks)
4 MARKS
17. Simplify each algebraic expression, leaving your answer in index form.
(a) 6x⁴ × 2x³ (1 mark)
(b) 18a⁶b⁴ ÷ 6a²b (2 marks)
(c) (3m²)³ (2 marks)
5 MARKS
18. Work with scientific notation.
(a) Write 0.00082 in scientific notation. (1 mark)
(b) Write 4.7 × 10⁶ as an ordinary number. (1 mark)
(c) Calculate (6 × 10⁷) ÷ (3 × 10²), giving your answer in scientific notation. (2 marks)
4 MARKS
Extended Response
19. This question brings together the index laws, scientific notation and fractional indices.
(a) Simplify (2a³)² × 3a⁴ ÷ 4a⁵, giving your answer with a positive index. Show each stage (power-of-a-power, then product, then quotient). (3 marks)
(b) Evaluate the fractional index 16^(3/4). (2 marks)
(c) A virus is about 3 × 10⁻⁷ m wide and a human cell is about 2 × 10⁻⁵ m wide. How many times wider is the cell than the virus? Give your answer in scientific notation, to 2 significant figures. (2 marks)
(d) Explain how the rule for multiplying powers of the same base (add the indices) is used when you multiply two numbers written in scientific notation. (1 mark)
8 MARKS
1. B, The base is 6 and the index (power) is 5; 6⁵ means five factors of 6.
2. C, 3³ = 3 × 3 × 3 = 27.
3. A, Product rule: same base, add indices. 4² × 4⁵ = 4²⁺⁵ = 4⁷.
4. D, Quotient rule: same base, subtract indices. 9⁶ ÷ 9² = 9⁶⁻² = 9⁴.
5. B, Power of a power: multiply indices. (5³)² = 5³ˣ² = 5⁶.
6. C, 2⁶ × 2² ÷ 2³ = 2⁶⁺²⁻³ = 2⁵.
7. A, Any non-zero number to the power 0 is 1, so 8⁰ = 1.
8. B, A negative index gives the reciprocal: 4⁻¹ = 1/4¹ = 1/4.
9. D, 2⁻³ = 1/2³ = 1/8.
10. C, Multiply coefficients (3 × 4 = 12) and add indices (x²⁺³): 12x⁵.
11. A, Divide coefficients (20 ÷ 5 = 4) and subtract indices (a⁷⁻³): 4a⁴.
12. B, Square the coefficient and multiply the index by 2: (2y³)² = 2²y³ˣ² = 4y⁶.
13. D, 0.00056 = 5.6 × 10⁻⁴ (decimal moved 4 places; small number → negative power; front number must be between 1 and 10).
14. A, Multiply the front numbers (3 × 2 = 6) and add the powers (10⁵⁺⁴ = 10⁹): 6 × 10⁹.
15. C, 25^(1/2) means the square root of 25, which is 5.
16 (4 marks): (a) 5⁴ × 5² ÷ 5³ = 5⁴⁺²⁻³ = 5³ [2]. (b) 2⁻² = 1/4 and 3⁰ = 1, so 1/4 + 1 = 1/4 + 4/4 = 5/4 [2].
17 (5 marks): (a) 6x⁴ × 2x³ = (6 × 2)x⁴⁺³ = 12x⁷ [1]. (b) 18a⁶b⁴ ÷ 6a²b = (18 ÷ 6)a⁶⁻²b⁴⁻¹ = 3a⁴b³ [2]. (c) (3m²)³ = 3³m²ˣ³ = 27m⁶ [2].
18 (4 marks): (a) 0.00082 = 8.2 × 10⁻⁴ [1]. (b) 4.7 × 10⁶ = 4,700,000 [1]. (c) (6 ÷ 3) × 10⁷⁻² = 2 × 10⁵ [2].
19 (8 marks):
(a) Power-of-a-power: (2a³)² = 4a⁶ [1]. Product rule: 4a⁶ × 3a⁴ = 12a¹⁰ [1]. Quotient rule: 12a¹⁰ ÷ 4a⁵ = (12 ÷ 4)a¹⁰⁻⁵ = 3a⁵ [1].
(b) 16^(3/4) = (⁴√16)³ = 2³ = 8 [2].
(c) (2 × 10⁻⁵) ÷ (3 × 10⁻⁷) = (2 ÷ 3) × 10⁻⁵⁻⁽⁻⁷⁾ = 0.667 × 10² = 66.7 = 6.7 × 10¹ (2 s.f.), the cell is about 67 times wider [2].
(d) When multiplying numbers in scientific notation you multiply the front numbers and multiply the powers of 10; because the powers of 10 share the base 10, the product rule applies and you simply add the indices (e.g. 10⁵ × 10⁴ = 10⁵⁺⁴ = 10⁹) [1].