Lesson 9: Negative Exponents and the Laws of Exponents

What does a negative exponent mean?

Rule:
x⁻ⁿ = 1 / xⁿ, for x ≠ 0

It means:

• x⁻¹ = 1/x
• x⁻² = 1/x²
• x⁻³ = 1/x³

Negative exponents do not mean the answer is negative.
They mean we take the reciprocal (flip the fraction).

Exercises

1. Verify the rule x⁻ⁿ = 1/xⁿ for x = 3 and n = 5.
Solution:
x⁻⁵ = 3⁻⁵ = 1/3⁵ = 1/243

2. What is (3 × 10⁻²)?
Solution:
10⁻² = 1/10² = 1/100 = 0.01
So (3 × 10⁻²) = 3 × 0.01 = 0.03

3. What is (3 × 10⁻⁵)?
Solution:
10⁻⁵ = 1/100000 = 0.00001
So (3 × 10⁻⁵) = 0.00003

4. Write 4.728 in expanded form using powers of 10.
Solution:
4.728 = 4×10⁰ + 7×10⁻¹ + 2×10⁻² + 8×10⁻³

5. 5³ = ?
Answer: 5³ = 125

6. 1/8⁹ = ?
Solution: 1/8⁹ = 8⁻⁹

7. 3 × 2⁻⁴ = ?
Solution: 2⁻⁴ = 1/2⁴ = 1/16
So 3 × 1/16 = 3/16

8. x⁻³ = ?
Answer: 1/x³

9. 1/x⁹ = ?
Answer: x⁻⁹

10. (x/y)⁻⁴ = ?
Solution: (x/y)⁻⁴ = (y/x)⁴

11. 19² ÷ 19⁵ = ?
Solution: Subtract exponents: 2 − 5 = −3
Answer: 19⁻³ = 1/19³

12. 17¹⁶ ÷ 17³ = ?
Solution: Subtract exponents: 16 − 3 = 13
Answer: 17¹³

13. Show that (7/5)⁻⁴ = 7⁻⁴ / 5⁻⁴.
Solution: (7/5)⁻⁴ = (5/7)⁴ = 5⁴/7⁴.
Also, 7⁻⁴/5⁻⁴ = (1/7⁴)/(1/5⁴) = 5⁴/7⁴. Both are equal.

14. Compute: 3³ × 3² × 3¹ × 3⁰ × 3⁻¹ × 3⁻²
Solution: Add exponents: 3+2+1+0+(−1)+(−2) = 3
Answer: 3³ = 27

15. Compute: 5² × 5¹⁰ × 5⁸ × 5⁰ × 5⁻¹⁰ × 5⁻⁸
Solution: Add exponents: 2+10+8+0+(−10)+(−8) = 2
Answer: 5² = 25

16. General: aᵐ × aⁿ × a⁰ × a⁻ⁿ × a⁻ᵐ = ?
Solution: Add exponents: m+n+0+(−n)+(−m) = 0
Answer: a⁰ = 1

17. Without using the rule, show that (17.6⁻¹)⁸ = 17.6⁻⁸.
Solution: (17.6⁻¹)⁸ = (1/17.6)⁸ = 1/(17.6)⁸ = 17.6⁻⁸

18. Without using the rule, prove (x⁻¹)ⁿ = x⁻ⁿ.
Solution: (x⁻¹)ⁿ = (1/x)ⁿ = 1/xⁿ = x⁻ⁿ

19. Without using rules, show that (2.8/2.8)ⁿ = 2.8⁰ = 1.
Solution: Numerator and denominator cancel → always 1.

20. 7⁻² = ?
Answer: 1/7² = 1/49

21. (1/5)⁻³ = ?
Answer: 5³ = 125

22. (−2)⁻⁴ = ?
Answer: 1/(−2)⁴ = 1/16

23. 10⁻⁶ = ?
Answer: 1/1,000,000 = 0.000001

24. x⁵ × x⁻⁵ = ?
Answer: x⁰ = 1

25. (y⁻³)(y²) = ?
Answer: y⁻¹ = 1/y

26. 4⁻² × 4³ = ?
Answer: 4¹ = 4

27. (ab)⁻² = ?
Answer: 1/(a²b²)

28. (x²/y³)⁻¹ = ?
Answer: y³/x²

29. Simplify: (2⁻³)(3⁻²)
Answer: 1/(2³×3²) = 1/72

30. Simplify: (5⁻¹)³
Answer: 5⁻³ = 1/125