Math worksheet on exponent rules and negative exponents.

I'll solve each problem step by step, applying the exponent rules carefully.

Key Rules to Remember:
- When multiplying same bases: add exponents (a^m · a^n = a^(m+n))
- Negative exponent means reciprocal (a^(-n) = 1/a^n)
- Power of a power: multiply exponents ((a^m)^n = a^(m·n))
- When multiplying coefficients, multiply them separately

Let me work through each problem:

1. 5³ = 5 × 5 × 5 = 125

2. 10³ = 10 × 10 × 10 = 1000

3. 4x⁻⁶y = 4y/x⁶ (move x⁻⁶ to denominator to make it positive) = 4y/x⁶

4. 8¹ · 8⁴ = 8^(1+4) = 8⁵

5. 7² · 7 = 7^(2+7) = 7⁹

6. x⁵ · x⁴ = x^(5+4) = x⁹

7. 2³ · 2⁻⁶ = 2^(3+(-6)) = 2^(-3) = 1/2³ or 1/8

8. 8¹ · 8³ = 8^(1+3) = 8⁴

9. t⁴ · t⁻ = t^(4+(-4)) = t⁰ = 1

10. x⁵ · x⁻⁹ = x^(5+(-9)) = x^(-4) = 1/x⁴

11. 3⁴ · x³ · x⁵ = 3⁴ · x^(3+5) = 3⁴ · x⁸ = 81x⁸

12. (6x²)(4x²) = (6·4)(x²·x²) = 24x^(2+2) = 24x⁴

13. (3x³y⁻²)(-6y⁵) = (3·-6)(x³)(y⁻²·y⁵) = -18x³y^(-2+5) = -18x³y³

14. (5p²)(-m⁸p²) = (5·-1)(m⁸)(p²·p²) = -5m⁸p^(2+2) = -5m⁸p⁴

15. (4fh⁻¹³)(-5f⁻⁶)(-3h³)
= (4·-5·-3)(f⁶·f⁻⁶)(h⁻¹³·h³)
= 60·f^(6+(-6))·h^(-13+3)
= 60·f⁰·h⁻¹⁰
= 60·1·h⁻¹⁰
= 60/h¹⁰

16. (-2²x³y⁴)((-3)²x⁴y⁻⁴)
First simplify: -2² = -4 and (-3)² = 9
= (-4x³y⁴)(9x⁴y⁻⁴)
= (-4·9)(x³·x⁴)(y⁴·y⁻⁴)
= -36·x^(3+4)·y^(4+(-4))
= -36·x⁷·y⁰
= -36·x⁷·1
= -36x⁷

17. (p²) = p^(2·5) = p¹⁰

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Final Answer:
1. 125
2. 1000
3. 4y/x⁶
4. 8⁵
5. 7⁹
6. x⁹
7. 1/8 (or 1/2³)
8. 8⁴
9. 1
10. 1/x⁴
11. 81x⁸
12. 24x⁴
13. -18x³y³
14. -5m⁸p⁴
15. 60/h¹⁰
16. -36x⁷
17. p¹⁰