Exponents Worksheet for solving expressions with various bases and exponents.

Certainly! Let's solve each problem on the Exponents Worksheet step by step, explaining the rules of exponents as we go.

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📌 Key Exponent Rules Used:

1. Zero Exponent Rule: Any non-zero number raised to the power of 0 is 1.
> \( a^0 = 1 \) (for \( a \ne 0 \))

2. Negative Exponent Rule: A negative exponent means take the reciprocal of the base and make the exponent positive.
> \( a^{-n} = \frac{1}{a^n} \)

3. Fractional Base with Negative Exponent: Apply the negative exponent rule — flip the fraction and make exponent positive.
> \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n \)

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## ✔ Solutions:

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1 a. \( 0.6^{-2} \)

→ Use negative exponent rule:
\( 0.6^{-2} = \frac{1}{0.6^2} \)

First, compute \( 0.6^2 = 0.36 \)

So, \( \frac{1}{0.36} = \frac{100}{36} = \frac{25}{9} \approx 2.777... \)

✔ Answer: \( \frac{25}{9} \) or approximately 2.78

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1 b. \( \left(\frac{1}{8}\right)^0 \)

→ Zero exponent rule: any non-zero base to the 0 power is 1.

✔ Answer: 1

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2 a. \( 9^{-1} \)

→ Negative exponent: \( \frac{1}{9^1} = \frac{1}{9} \)

✔ Answer: \( \frac{1}{9} \)

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2 b. \( 1^0 \)

→ Zero exponent rule: \( 1^0 = 1 \)

✔ Answer: 1

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3 a. \( \left(\frac{1}{5}\right)^0 \)

→ Again, zero exponent: any non-zero base to 0 is 1.

✔ Answer: 1

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3 b. \( 0.2^{-3} \)

→ First, note that \( 0.2 = \frac{1}{5} \), so:

\( 0.2^{-3} = \left(\frac{1}{5}\right)^{-3} = 5^3 = 125 \)

✔ Answer: 125

*(Alternatively: \( 0.2^{-3} = \frac{1}{0.2^3} = \frac{1}{0.008} = 125 \))*

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4 a. \( 0.4^0 \)

→ Zero exponent rule: \( 0.4^0 = 1 \)

✔ Answer: 1

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4 b. \( 7^{-2} \)

→ Negative exponent: \( \frac{1}{7^2} = \frac{1}{49} \)

✔ Answer: \( \frac{1}{49} \)

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5 a. \( \left(\frac{1}{2}\right)^{-1} \)

→ Flip the fraction: \( \left(\frac{2}{1}\right)^1 = 2 \)

✔ Answer: 2

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5 b. \( 10^{-8} \)

→ Negative exponent: \( \frac{1}{10^8} = \frac{1}{100,000,000} \)

✔ Answer: \( \frac{1}{100000000} \) or 0.00000001

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6 a. \( \left(\frac{4}{5}\right)^0 \)

→ Zero exponent: any non-zero base to 0 is 1.

✔ Answer: 1

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6 b. \( \left(\frac{1}{2}\right)^{-2} \)

→ Flip the fraction and make exponent positive:
\( \left(\frac{2}{1}\right)^2 = 2^2 = 4 \)

✔ Answer: 4

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7 a. \( 0.3^{-3} \)

→ \( 0.3 = \frac{3}{10} \), so:

\( \left(\frac{3}{10}\right)^{-3} = \left(\frac{10}{3}\right)^3 = \frac{1000}{27} \approx 37.037 \)

✔ Answer: \( \frac{1000}{27} \) or approximately 37.04

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7 b. \( 0.5^0 \)

→ Zero exponent: \( 0.5^0 = 1 \)

✔ Answer: 1

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## 🧾 Final Answers Summary:

| Problem | Answer |
|---------|-----------------|
| 1a | \( \frac{25}{9} \) |
| 1b | 1 |
| 2a | \( \frac{1}{9} \) |
| 2b | 1 |
| 3a | 1 |
| 3b | 125 |
| 4a | 1 |
| 4b | \( \frac{1}{49} \) |
| 5a | 2 |
| 5b | \( \frac{1}{100000000} \) |
| 6a | 1 |
| 6b | 4 |
| 7a | \( \frac{1000}{27} \) |
| 7b | 1 |

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Let me know if you’d like these answers in decimal form or simplified further! 😊