This checkpoint assesses your understanding of the zero index, negative indices (writing them with positive indices and evaluating them), and combining all the numerical index laws. Show all working for short answer questions.
Multiple Choice
1. Evaluate 7⁰.
2. Which statement about the zero index is correct?
3. Write 2⁻¹ as a fraction.
4. Rewrite 5⁻² with a positive index.
5. Evaluate 2⁻³ as a fraction.
6. Evaluate 4³ × 7⁰.
7. Simplify a⁵ ÷ a⁵ using the quotient rule.
8. Simplify 2⁴ × 2⁻⁶, giving your answer with a positive index.
9. Evaluate 3⁻² + 2⁰.
10. Simplify (6⁵ ÷ 6³) × 6⁻², leaving your answer as a single power of 6.
Short Answer
11. Evaluate each expression, giving your answer as a whole number or fraction.
(a) 9⁰ (1 mark)
(b) 5 × 8⁰ (1 mark)
(c) 3⁻² (1 mark)
3 MARKS
12. Rewrite each expression using only positive indices, then evaluate where possible.
(a) 4⁻² (1 mark)
(b) x⁻⁵ (1 mark)
(c) 2⁻⁴ (give the value as a fraction) (1 mark)
(d) Explain in one sentence what a negative index tells you to do. (1 mark)
4 MARKS
13. Consider the expression a⁷ ÷ a¹⁰.
(a) Use the quotient rule (subtract the indices) to write the result with a single index. (1 mark)
(b) Rewrite your answer from (a) with a positive index. (1 mark)
(c) Show how the same result is found by writing the division as a fraction and cancelling. (1 mark)
3 MARKS
14. Simplify the expression 2⁵ × 2⁻³ ÷ 2⁴, using the numerical index laws.
(a) Use the product rule on 2⁵ × 2⁻³. (1 mark)
(b) Apply the quotient rule by dividing by 2⁴. (1 mark)
(c) Write the result as a single power of 2. (1 mark)
(d) Rewrite your answer with a positive index and evaluate it as a fraction. (1 mark)
(e) Explain why combining the indices first (before evaluating) makes the calculation easier. (1 mark)
5 MARKS
1. C, Any non-zero number to the power 0 equals 1, so 7⁰ = 1.
2. B, The zero index rule is a⁰ = 1 for any non-zero a (it comes from aⁿ ÷ aⁿ = a⁰ = 1).
3. A, A negative index gives the reciprocal: 2⁻¹ = 1/2¹ = 1/2.
4. C, 5⁻² = 1/5² (the negative index makes it a reciprocal; the index itself becomes positive).
5. D, 2⁻³ = 1/2³ = 1/8.
6. B, 7⁰ = 1, so 4³ × 7⁰ = 64 × 1 = 64.
7. A, Quotient rule: a⁵ ÷ a⁵ = a⁵⁻⁵ = a⁰ = 1.
8. C, Product rule: 2⁴ × 2⁻⁶ = 2⁴⁺⁽⁻⁶⁾ = 2⁻² = 1/2².
9. B, 3⁻² = 1/9 and 2⁰ = 1, so 1/9 + 1 = 1/9 + 9/9 = 10/9.
10. D, 6⁵ ÷ 6³ = 6²; 6² × 6⁻² = 6²⁺⁽⁻²⁾ = 6⁰ (which equals 1).
11 (3 marks): (a) 9⁰ = 1 [1]. (b) 5 × 8⁰ = 5 × 1 = 5 [1]. (c) 3⁻² = 1/3² = 1/9 [1].
12 (4 marks): (a) 4⁻² = 1/4² = 1/16 [1]. (b) x⁻⁵ = 1/x⁵ [1]. (c) 2⁻⁴ = 1/2⁴ = 1/16 [1]. (d) A negative index means take the reciprocal of the base raised to the matching positive index (i.e. a⁻ⁿ = 1/aⁿ) [1].
13 (3 marks): (a) a⁷ ÷ a¹⁰ = a⁷⁻¹⁰ = a⁻³ [1]. (b) a⁻³ = 1/a³ [1]. (c) a⁷/a¹⁰ = (a·a·a·a·a·a·a)/(a·a·a·a·a·a·a·a·a·a); cancel the seven common factors to leave 1/(a·a·a) = 1/a³ [1].
14 (5 marks): (a) 2⁵ × 2⁻³ = 2⁵⁺⁽⁻³⁾ = 2² [1]. (b) 2² ÷ 2⁴ = 2²⁻⁴ [1]. (c) = 2⁻² [1]. (d) 2⁻² = 1/2² = 1/4 [1]. (e) Combining the indices first reduces the whole expression to one small power of 2, so you only evaluate a single power instead of three separate large numbers [1].