Lesson 8: Numbers Raised to the Zeroth Power

What happens when any number (except 0) is raised to the power of 0?

Rule:
x⁰ = 1, for all x ≠ 0

Why?
Think about division with exponents:
x³ ÷ x³ = x³⁻³ = x⁰
But x³ ÷ x³ = 1
So x⁰ must equal 1.

Exercises

1. List all possible cases of whole numbers m and n for the rule xᵐ × xⁿ = xᵐ⁺ⁿ.
Solution:
We know it works for m > 0, n > 0.
We still need to test cases where one exponent = 0.

2. Check the rule xᵐ × xⁿ = xᵐ⁺ⁿ for m = 0, n > 0.
Example: x⁰ × x³ = ?
x⁰ = 1, so 1 × x³ = x³.
Right side: x⁰⁺³ = x³. Works.

3. Check the rule (xᵐ)ⁿ = xᵐⁿ when one exponent is 0.
Example: (x³)⁰ = ?
(x³)⁰ = 1
Right side: x³×⁰ = x⁰ = 1. Works.

4. Check the rule (x/y)ⁿ = xⁿ/yⁿ when n = 0.
Example: (5/7)⁰ = ?
= 1
Right side: 5⁰ / 7⁰ = 1/1 = 1. Works.

5. Write the expanded form of 8,374 using exponential notation.
Solution:
8,374 = 8×10³ + 3×10² + 7×10¹ + 4×10⁰

6. Write the expanded form of 6,985,062 using exponential notation.
Solution:
6,985,062 = 6×10⁶ + 9×10⁵ + 8×10⁴ + 5×10³ + 0×10² + 6×10¹ + 2×10⁰

7. 12⁰ ÷ 12⁰ = ?
Solution: Both = 1 → 1 ÷ 1 = 1

8. 9¹⁵ × 9⁻¹⁵ = ?
Solution: Subtract exponents: 15 − 15 = 0 → 9⁰ = 1

9. (7(123456.789)⁴)⁰ = ?
Solution: Anything nonzero to power 0 = 1.
Answer: 1

10. 2² × 2⁻⁵ × 2⁵ × 2⁻² = ?
Solution: Add exponents: 2 + (−5) + 5 + (−2) = 0
So answer = 2⁰ = 1

11. x⁴¹ ÷ x⁴¹ = ?
Solution: Subtract exponents: 41 − 41 = 0 → x⁰ = 1 (x ≠ 0)

12. 5⁰ = ?
Answer: 1

13. (−7)⁰ = ?
Answer: 1

14. (3/4)⁰ = ?
Answer: 1

15. 100⁰ = ?
Answer: 1

16. (2x)⁰ = ?
Answer: 1 (as long as x ≠ 0)

17. x⁰y⁰ = ?
Answer: 1×1 = 1

18. (10²)⁰ = ?
Answer: 1

19. (a²b³c⁴)⁰ = ?
Answer: 1

20. 7³ ÷ 7³ = ?
Solution: Subtract exponents → 7⁰ = 1

21. 2⁵ × 2⁻⁵ = ?
Solution: Add exponents: 5 + (−5) = 0 → 2⁰ = 1

22. (−3)⁴ ÷ (−3)⁴ = ?
Solution: (−3)⁰ = 1

23. Simplify: (x²y³)⁰
Answer: 1

24. Simplify: (5⁴ × 2³)⁰
Answer: 1