Worksheet on simplifying expressions with negative and zero exponents.

Let's solve the worksheet on Negative Exponents and Zero Exponents step by step. I'll explain the rules first, then simplify each expression.

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🔹 Key Rules:

1. Negative Exponent Property:
$$
x^{-n} = \frac{1}{x^n}
$$
This means a negative exponent indicates the reciprocal of the base raised to the positive exponent.

2. Zero Exponent Property:
$$
x^0 = 1
$$
Any non-zero number raised to the power of zero is 1.

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Now, let's go through each problem:

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✔ Simplify using the Negative Exponent Property

#### 1. $ 3^{-1} $
$$
3^{-1} = \frac{1}{3^1} = \frac{1}{3}
$$

#### 2. $ 2^{-4} $
$$
2^{-4} = \frac{1}{2^4} = \frac{1}{16}
$$

#### 3. $ 6^{-1} $
$$
6^{-1} = \frac{1}{6}
$$

#### 4. $ 4^{-2} $
$$
4^{-2} = \frac{1}{4^2} = \frac{1}{16}
$$

#### 5. $ 13^{-1} $
$$
13^{-1} = \frac{1}{13}
$$

#### 6. $ 9^{-2} $
$$
9^{-2} = \frac{1}{9^2} = \frac{1}{81}
$$

#### 7. $ 12^{-2} $
$$
12^{-2} = \frac{1}{12^2} = \frac{1}{144}
$$

#### 8. $ 1^{-1} $
$$
1^{-1} = \frac{1}{1^1} = \frac{1}{1} = 1
$$

#### 9. $ 22^{-1} $
$$
22^{-1} = \frac{1}{22}
$$

#### 10. $ 19^{-1} $
$$
19^{-1} = \frac{1}{19}
$$

#### 11. $ 13^{-3} $
$$
13^{-3} = \frac{1}{13^3} = \frac{1}{2197}
$$

#### 12. $ 250^{-1} $
$$
250^{-1} = \frac{1}{250}
$$

#### 13. $ 0.8^{-1} $
$$
0.8^{-1} = \frac{1}{0.8} = \frac{1}{\frac{4}{5}} = \frac{5}{4} = 1.25
$$

#### 14. $ 20^{-1} $
$$
20^{-1} = \frac{1}{20}
$$

#### 15. $ 1.6^{-1} $
$$
1.6^{-1} = \frac{1}{1.6} = \frac{1}{\frac{8}{5}} = \frac{5}{8} = 0.625
$$

#### 16. $ 5^{-3} $
$$
5^{-3} = \frac{1}{5^3} = \frac{1}{125}
$$

#### 17. $ \left(\frac{2}{3}\right)^{-1} $
$$
\left(\frac{2}{3}\right)^{-1} = \frac{1}{\frac{2}{3}} = \frac{3}{2}
$$

#### 18. $ 77^{-1} $
$$
77^{-1} = \frac{1}{77}
$$

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✔ Final Answers (Fill in the blanks):

| Expression | Simplified |
|-----------|------------|
| $ 3^{-1} $ | $ \frac{1}{3} $ |
| $ 2^{-4} $ | $ \frac{1}{16} $ |
| $ 6^{-1} $ | $ \frac{1}{6} $ |
| $ 4^{-2} $ | $ \frac{1}{16} $ |
| $ 13^{-1} $ | $ \frac{1}{13} $ |
| $ 9^{-2} $ | $ \frac{1}{81} $ |
| $ 12^{-2} $ | $ \frac{1}{144} $ |
| $ 1^{-1} $ | $ 1 $ |
| $ 22^{-1} $ | $ \frac{1}{22} $ |
| $ 19^{-1} $ | $ \frac{1}{19} $ |
| $ 13^{-3} $ | $ \frac{1}{2197} $ |
| $ 250^{-1} $ | $ \frac{1}{250} $ |
| $ 0.8^{-1} $ | $ \frac{5}{4} $ or $ 1.25 $ |
| $ 20^{-1} $ | $ \frac{1}{20} $ |
| $ 1.6^{-1} $ | $ \frac{5}{8} $ or $ 0.625 $ |
| $ 5^{-3} $ | $ \frac{1}{125} $ |
| $ \left(\frac{2}{3}\right)^{-1} $ | $ \frac{3}{2} $ |
| $ 77^{-1} $ | $ \frac{1}{77} $ |

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📌 Summary of Key Ideas:

- A negative exponent flips the base into the denominator.
- The zero exponent always gives 1 (as long as the base is not zero).
- For fractions with negative exponents: $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $

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