This checkpoint assesses your understanding of index notation (bases and powers), evaluating powers, and the product, quotient and power-of-a-power laws, including combining them in numerical expressions. Show all working for short answer questions.
Multiple Choice
1. In the expression 3⁴, which number is the base?
2. Evaluate 2⁵.
3. Which expression is equal to 4 × 4 × 4 × 4 × 4 × 4?
4. Simplify 5³ × 5² using the product rule.
5. Simplify 7⁸ ÷ 7³ using the quotient rule.
6. Simplify (4²)³ using the power-of-a-power rule.
7. Evaluate 10³ × 10² as an ordinary number.
8. Simplify 2³ × 2⁴ ÷ 2² and give the answer as a number.
9. Which expression is equal to 3⁶?
10. Simplify (5²)³ × 5⁴ ÷ 5⁸, leaving your answer as a single power of 5.
Short Answer
11. Evaluate each power as an ordinary number.
(a) 3⁴ (1 mark)
(b) 2⁶ (1 mark)
(c) 10⁵ (1 mark)
3 MARKS
12. Use the index laws to simplify each expression, leaving your answer as a single power.
(a) 6⁵ × 6³ (1 mark)
(b) 9⁷ ÷ 9⁴ (1 mark)
(c) (2⁴)² (1 mark)
(d) Explain in one sentence why 4³ × 5² cannot be written as a single power. (1 mark)
4 MARKS
13. Simplify the expression 2³ × 2⁵ ÷ 2⁴.
(a) Use the product rule to simplify the numerator 2³ × 2⁵. (1 mark)
(b) Apply the quotient rule by dividing by 2⁴. (1 mark)
(c) Write the final answer as a single power of 2. (1 mark)
(d) Evaluate that power as an ordinary number. (1 mark)
4 MARKS
1. A, In 3⁴ the base is the number being multiplied repeatedly: 3. The 4 is the index (power).
2. C, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
3. B, Six factors of 4 multiplied together is written 4⁶ (base 4, index 6).
4. D, Product rule: same base, add indices. 5³ × 5² = 5³⁺² = 5⁵. The base stays 5; do not multiply the bases or the indices.
5. B, Quotient rule: same base, subtract indices. 7⁸ ÷ 7³ = 7⁸⁻³ = 7⁵.
6. C, Power of a power: multiply the indices. (4²)³ = 4²ˣ³ = 4⁶.
7. A, 10³ × 10² = 10³⁺² = 10⁵ = 100,000.
8. B, 2³ × 2⁴ ÷ 2² = 2³⁺⁴⁻² = 2⁵ = 32.
9. C, (3²)³ = 3²ˣ³ = 3⁶. (Check: 3² × 3³ = 3⁵; 3⁸ ÷ 3⁴ = 3⁴; (3³)³ = 3⁹, none equal 3⁶.)
10. D, (5²)³ = 5⁶; 5⁶ × 5⁴ = 5¹⁰; 5¹⁰ ÷ 5⁸ = 5².
11 (3 marks): (a) 3⁴ = 3 × 3 × 3 × 3 = 81 [1]. (b) 2⁶ = 64 [1]. (c) 10⁵ = 100,000 [1].
12 (4 marks): (a) 6⁵ × 6³ = 6⁵⁺³ = 6⁸ [1]. (b) 9⁷ ÷ 9⁴ = 9⁷⁻⁴ = 9³ [1]. (c) (2⁴)² = 2⁴ˣ² = 2⁸ [1]. (d) The index laws for multiplying only combine powers with the same base; 4³ × 5² has different bases (4 and 5), so the indices cannot be added [1].
13 (4 marks): (a) 2³ × 2⁵ = 2³⁺⁵ = 2⁸ [1]. (b) 2⁸ ÷ 2⁴ = 2⁸⁻⁴ [1]. (c) = 2⁴ [1]. (d) 2⁴ = 16 [1].