Exponents Worksheet featuring ten math problems to solve, including operations with fractions, decimals, and powers.

Here's the complete solution to the Exponents Worksheet, with step-by-step explanations for each problem.

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1 a. $\left(\frac{2}{3}\right)^2 \div \left(\frac{1}{2}\right)^6$

Step 1: Compute each exponent separately.

- $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$
- $\left(\frac{1}{2}\right)^6 = \frac{1^6}{2^6} = \frac{1}{64}$

Step 2: Divide the two fractions:

$$
\frac{4}{9} \div \frac{1}{64} = \frac{4}{9} \times \frac{64}{1} = \frac{256}{9}
$$

✔ Answer: $\boxed{\dfrac{256}{9}}$

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1 b. $\left(\frac{5}{7}\right)^2 \times \left(\frac{5}{7}\right)^1$

Step 1: When multiplying powers with the same base, add the exponents.

$$
\left(\frac{5}{7}\right)^{2+1} = \left(\frac{5}{7}\right)^3
$$

Step 2: Compute the cube.

$$
\left(\frac{5}{7}\right)^3 = \frac{5^3}{7^3} = \frac{125}{343}
$$

✔ Answer: $\boxed{\dfrac{125}{343}}$

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2 a. $0.6^1 \times 0.2^3$

Step 1: Compute each part.

- $0.6^1 = 0.6$
- $0.2^3 = 0.2 \times 0.2 \times 0.2 = 0.008$

Step 2: Multiply.

$$
0.6 \times 0.008 = 0.0048
$$

✔ Answer: $\boxed{0.0048}$

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2 b. $\left(\frac{2}{5}\right)^3 \times \left(\frac{2}{5}\right)^2$

Step 1: Add exponents (same base).

$$
\left(\frac{2}{5}\right)^{3+2} = \left(\frac{2}{5}\right)^5
$$

Step 2: Compute the fifth power.

$$
\left(\frac{2}{5}\right)^5 = \frac{2^5}{5^5} = \frac{32}{3125}
$$

✔ Answer: $\boxed{\dfrac{32}{3125}}$

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3 a. $1^{99} - 0.6^2$

Step 1: Any power of 1 is 1.

- $1^{99} = 1$

Step 2: Compute $0.6^2 = 0.36$

Step 3: Subtract.

$$
1 - 0.36 = 0.64
$$

✔ Answer: $\boxed{0.64}$

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3 b. $0.2^1 \times \left(\frac{1}{8}\right)^2$

Step 1: Compute each part.

- $0.2^1 = 0.2$
- $\left(\frac{1}{8}\right)^2 = \frac{1}{64}$

Step 2: Multiply.

$$
0.2 \times \frac{1}{64} = \frac{2}{10} \times \frac{1}{64} = \frac{2}{640} = \frac{1}{320}
$$

(Alternatively, as decimal: $0.2 \div 64 = 0.003125$)

✔ Answer: $\boxed{\dfrac{1}{320}}$ or $\boxed{0.003125}$

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4 a. $\left(\frac{1}{2}\right)^2 \div \left(\frac{5}{8}\right)^1$

Step 1: Compute each part.

- $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$
- $\left(\frac{5}{8}\right)^1 = \frac{5}{8}$

Step 2: Divide.

$$
\frac{1}{4} \div \frac{5}{8} = \frac{1}{4} \times \frac{8}{5} = \frac{8}{20} = \frac{2}{5}
$$

✔ Answer: $\boxed{\dfrac{2}{5}}$

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4 b. $7^2 - 0.5^3$

Step 1: Compute each part.

- $7^2 = 49$
- $0.5^3 = 0.125$

Step 2: Subtract.

$$
49 - 0.125 = 48.875
$$

✔ Answer: $\boxed{48.875}$

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5 a. $3^1 \div 0.8^2$

Step 1: Compute each part.

- $3^1 = 3$
- $0.8^2 = 0.64$

Step 2: Divide.

$$
3 \div 0.64 = \frac{3}{0.64} = \frac{300}{64} = \frac{75}{16} = 4.6875
$$

✔ Answer: $\boxed{4.6875}$ or $\boxed{\dfrac{75}{16}}$

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5 b. $0.7^2 + 0.9^1$

Step 1: Compute each part.

- $0.7^2 = 0.49$
- $0.9^1 = 0.9$

Step 2: Add.

$$
0.49 + 0.9 = 1.39
$$

✔ Answer: $\boxed{1.39}$

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## ✔ Final Answers Summary:

| Problem | Answer |
|---------|---------------------|
| 1a | $\dfrac{256}{9}$ |
| 1b | $\dfrac{125}{343}$ |
| 2a | $0.0048$ |
| 2b | $\dfrac{32}{3125}$ |
| 3a | $0.64$ |
| 3b | $\dfrac{1}{320}$ or $0.003125$ |
| 4a | $\dfrac{2}{5}$ |
| 4b | $48.875$ |
| 5a | $4.6875$ or $\dfrac{75}{16}$ |
| 5b | $1.39$ |

Let me know if you’d like these answers in different formats (mixed numbers, decimals only, etc.)!