1. What is 25⁰?
2. Simplify 8⁵ ÷ 8⁵.
3. Simplify (2³)² × 2⁴.
4. Which statement has the necessary condition?
5. Evaluate 10⁰ + 3² × 2.
6. Use two different arguments to explain why 9⁰ = 1.
7. Simplify (5²)³ ÷ 5⁴.
8. Evaluate 2⁵ × 2² ÷ 2⁶ + 7⁰.
9. Create a numerical expression that uses all four Stage 4 index laws and evaluates to 17.
Answers and reasoning
1. Answer
1. Every non-zero base raised to zero equals 1.
2. Answer
1. 8⁵ ÷ 8⁵ = 8⁰ = 1.
3. Answer
2¹⁰. (2³)² = 2⁶, then 2⁶ × 2⁴ = 2¹⁰.
4. Answer
a⁰ = 1 for a ≠ 0. The Stage 4 zero-index law applies to non-zero bases.
5. Answer
19. 10⁰ = 1 and 3² × 2 = 18, so the total is 19.
6. Model response
Quotient law: 9³ ÷ 9³ = 9⁰, while any non-zero number divided by itself is 1. Pattern: 9² = 81, 9¹ = 9, and dividing by 9 again gives 9⁰ = 1.
7. Model response
5⁶ ÷ 5⁴ = 5² = 25.
8. Model response
2⁷ ÷ 2⁶ + 1 = 2¹ + 1 = 3.
9. Model response
Example: (2²)³ ÷ 2⁴ + 3⁰ + 2³ × 2⁰ = 2² + 1 + 8 = 13, not 17; revise to (2²)³ ÷ 2³ + 3⁰ + 2³ × 2⁰ = 8 + 1 + 8 = 17.