Math worksheet for evaluating exponents with fractions, suitable for classroom or home study.

Let's solve each of these exponent problems step by step. The key rules we’ll use:

- Positive exponents: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)
- Negative exponents: \( \left( \frac{a}{b} \right)^{-n} = \left( \frac{b}{a} \right)^n = \frac{b^n}{a^n} \)
- Even powers of negative numbers → positive result
- Odd powers of negative numbers → negative result

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

Simplify fraction first: \( -\frac{3}{12} = -\frac{1}{4} \)

Then square: \( \left(-\frac{1}{4}\right)^2 = \frac{1}{16} \)

✔ Answer: \( \frac{1}{16} \)

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

Square numerator and denominator: \( \frac{(-3)^2}{4^2} = \frac{9}{16} \)

✔ 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 the fraction and make exponent positive:

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

Now, odd power of negative → 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 base: \( \frac{5}{10} = \frac{1}{2} \)

So: \( \left(\frac{1}{2}\right)^{-2} = \left(\frac{2}{1}\right)^2 = 4 \)

✔ Answer: \( 4 \)

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

Simplify base: \( \frac{6}{8} = \frac{3}{4} \)

So: \( \left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3 = \frac{64}{27} \)

✔ Answer: \( \frac{64}{27} \)

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

Simplify: \( \frac{2}{6} = \frac{1}{3} \)

Then: \( \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)

✔ 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: \( \left(-\frac{2}{1}\right)^7 = (-2)^7 = -128 \)

✔ Answer: \( -128 \)

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

Simplify: \( \frac{2}{4} = \frac{1}{2} \)

Then: \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)

✔ Answer: \( \frac{1}{4} \)

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

Simplify: \( \frac{2}{12} = \frac{1}{6} \)

Then: \( \left(\frac{1}{6}\right)^2 = \frac{1}{36} \)

✔ 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 base: \( -\frac{3}{6} = -\frac{1}{2} \)

So: \( \left(-\frac{1}{2}\right)^{-2} = \left(-\frac{2}{1}\right)^2 = (-2)^2 = 4 \)

✔ Answer: \( 4 \)

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## ✔ Final Answers:

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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You can now fill in the worksheet with these answers! Let me know if you’d like a printable version or explanations for any specific problem.