Exponents: Printable Worksheets - Free Printable

Here is the complete solution to “Evaluate the Exponents” worksheet. Each problem involves evaluating a rational number raised to an integer exponent (positive, negative, or zero). I’ll show the step-by-step reasoning for each.

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🔢 Key Rules Used:

- Positive exponent: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)
- Negative exponent: \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n \)
- Even exponent on negative base: result is positive
- Odd exponent on negative base: result is negative
- Always simplify fractions when possible.

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## ✔ Solutions:

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1) \( \left(-\frac{3}{12}\right)^2 \)

→ Simplify inside: \( -\frac{3}{12} = -\frac{1}{4} \)
→ Square: \( \left(-\frac{1}{4}\right)^2 = \frac{1}{16} \)
✔ Answer: \( \frac{1}{16} \)

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2) \( \left(-\frac{3}{4}\right)^2 \)

→ Even power → positive
→ \( \frac{3^2}{4^2} = \frac{9}{16} \)
✔ Answer: \( \frac{9}{16} \)

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3) \( \left(\frac{1}{3}\right)^3 \)

→ \( \frac{1^3}{3^3} = \frac{1}{27} \)
✔ Answer: \( \frac{1}{27} \)

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4) \( \left(-\frac{4}{5}\right)^2 \)

→ Even power → positive
→ \( \frac{4^2}{5^2} = \frac{16}{25} \)
✔ Answer: \( \frac{16}{25} \)

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5) \( \left(-\frac{2}{3}\right)^{-5} \)

→ Negative exponent → flip fraction and make exponent positive
→ \( \left(-\frac{3}{2}\right)^5 \)
→ Odd power → negative
→ \( -\frac{3^5}{2^5} = -\frac{243}{32} \)
✔ Answer: \( -\frac{243}{32} \)

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6) \( \left(-\frac{5}{10}\right)^2 \)

→ Simplify: \( -\frac{5}{10} = -\frac{1}{2} \)
→ Square: \( \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \)
✔ Answer: \( \frac{1}{4} \)

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7) \( \left(-\frac{2}{7}\right)^3 \)

→ Odd power → negative
→ \( -\frac{2^3}{7^3} = -\frac{8}{343} \)
✔ Answer: \( -\frac{8}{343} \)

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8) \( \left(\frac{1}{2}\right)^6 \)

→ \( \frac{1^6}{2^6} = \frac{1}{64} \)
✔ Answer: \( \frac{1}{64} \)

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9) \( \left(\frac{5}{10}\right)^{-2} \)

→ Simplify: \( \frac{5}{10} = \frac{1}{2} \)
→ \( \left(\frac{1}{2}\right)^{-2} = \left(\frac{2}{1}\right)^2 = 4 \)
✔ Answer: \( 4 \)

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10) \( \left(\frac{6}{8}\right)^{-3} \)

→ Simplify: \( \frac{6}{8} = \frac{3}{4} \)
→ \( \left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3 = \frac{64}{27} \)
✔ Answer: \( \frac{64}{27} \)

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11) \( \left(\frac{2}{6}\right)^2 \)

→ Simplify: \( \frac{2}{6} = \frac{1}{3} \)
→ \( \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)
✔ Answer: \( \frac{1}{9} \)

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12) \( \left(-\frac{1}{2}\right)^8 \)

→ Even power → positive
→ \( \frac{1^8}{2^8} = \frac{1}{256} \)
✔ Answer: \( \frac{1}{256} \)

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13) \( \left(-\frac{2}{3}\right)^5 \)

→ Odd power → negative
→ \( -\frac{2^5}{3^5} = -\frac{32}{243} \)
✔ Answer: \( -\frac{32}{243} \)

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14) \( \left(-\frac{1}{2}\right)^{-7} \)

→ Negative exponent → flip and make positive
→ \( \left(-\frac{2}{1}\right)^7 = -128 \) (odd power → negative)
✔ Answer: \( -128 \)

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15) \( \left(\frac{2}{4}\right)^2 \)

→ Simplify: \( \frac{2}{4} = \frac{1}{2} \)
→ \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
✔ Answer: \( \frac{1}{4} \)

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16) \( \left(\frac{2}{12}\right)^2 \)

→ Simplify: \( \frac{2}{12} = \frac{1}{6} \)
→ \( \left(\frac{1}{6}\right)^2 = \frac{1}{36} \)
✔ Answer: \( \frac{1}{36} \)

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17) \( \left(-\frac{2}{9}\right)^2 \)

→ Even power → positive
→ \( \frac{2^2}{9^2} = \frac{4}{81} \)
✔ Answer: \( \frac{4}{81} \)

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18) \( \left(\frac{1}{2}\right)^6 \)

→ Same as #8: \( \frac{1}{64} \)
✔ Answer: \( \frac{1}{64} \)

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19) \( \left(\frac{1}{3}\right)^4 \)

→ \( \frac{1^4}{3^4} = \frac{1}{81} \)
✔ Answer: \( \frac{1}{81} \)

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20) \( \left(-\frac{3}{6}\right)^{-2} \)

→ Simplify: \( -\frac{3}{6} = -\frac{1}{2} \)
→ \( \left(-\frac{1}{2}\right)^{-2} = \left(-\frac{2}{1}\right)^2 = 4 \) (even power → positive)
✔ Answer: \( 4 \)

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## 📝 Final Answer Sheet:

| Problem | Answer |
|---------|------------------|
| 1 | \( \frac{1}{16} \) |
| 2 | \( \frac{9}{16} \) |
| 3 | \( \frac{1}{27} \) |
| 4 | \( \frac{16}{25} \) |
| 5 | \( -\frac{243}{32} \) |
| 6 | \( \frac{1}{4} \) |
| 7 | \( -\frac{8}{343} \) |
| 8 | \( \frac{1}{64} \) |
| 9 | \( 4 \) |
| 10 | \( \frac{64}{27} \) |
| 11 | \( \frac{1}{9} \) |
| 12 | \( \frac{1}{256} \) |
| 13 | \( -\frac{32}{243} \) |
| 14 | \( -128 \) |
| 15 | \( \frac{1}{4} \) |
| 16 | \( \frac{1}{36} \) |
| 17 | \( \frac{4}{81} \) |
| 18 | \( \frac{1}{64} \) |
| 19 | \( \frac{1}{81} \) |
| 20 | \( 4 \) |

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✔ All problems solved with simplification and correct application of exponent rules.

Let me know if you’d like a printable version or explanation of any specific problem!