Math worksheet for practicing negative exponents of fractions.

To solve these problems, we use the rule for negative exponents.

The Rule:
A negative exponent tells you to take the reciprocal of the base. This means you flip the fraction upside down and make the exponent positive.

Formula:
$$ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} $$

After flipping the fraction, you raise the new numerator and denominator to the power of the exponent. Often, it is easier to simplify the fraction inside the parentheses *before* flipping or calculating, but flipping first works too. Let's solve them row by row.

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Row 1

1. $\left(\frac{1}{5}\right)^{-2}$
* Flip the fraction: $\left(\frac{5}{1}\right)^{2}$
* Calculate: $5^2 = 25$
* Answer: 25

2. $\left(\frac{1}{3}\right)^{-2}$
* Flip the fraction: $\left(\frac{3}{1}\right)^{2}$
* Calculate: $3^2 = 9$
* Answer: 9

3. $\left(\frac{1}{4}\right)^{-3}$
* Flip the fraction: $\left(\frac{4}{1}\right)^{3}$
* Calculate: $4^3 = 4 \times 4 \times 4 = 64$
* Answer: 64

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Row 2

4. $\left(\frac{1}{7}\right)^{-2}$
* Flip the fraction: $\left(\frac{7}{1}\right)^{2}$
* Calculate: $7^2 = 49$
* Answer: 49

5. $\left(\frac{2}{6}\right)^{-2}$
* *Tip: Simplify first!* $\frac{2}{6}$ simplifies to $\frac{1}{3}$.
* Problem becomes: $\left(\frac{1}{3}\right)^{-2}$
* Flip the fraction: $\left(\frac{3}{1}\right)^{2}$
* Calculate: $3^2 = 9$
* Answer: 9
* *(Note: If you didn't simplify, you would get $(\frac{6}{2})^2 = 3^2 = 9$. Same result.)*

6. $\left(-\frac{1}{2}\right)^{-2}$
* Flip the fraction: $\left(-\frac{2}{1}\right)^{2}$ which is $(-2)^2$
* Calculate: $-2 \times -2 = 4$ (Negative times negative is positive)
* Answer: 4

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Row 3

7. $\left(\frac{1}{6}\right)^{-3}$
* Flip the fraction: $\left(\frac{6}{1}\right)^{3}$
* Calculate: $6^3 = 6 \times 6 \times 6 = 216$
* Answer: 216

8. $\left(\frac{1}{4}\right)^{-4}$
* Flip the fraction: $\left(\frac{4}{1}\right)^{4}$
* Calculate: $4^4 = 4 \times 4 \times 4 \times 4 = 256$
* Answer: 256

9. $\left(\frac{1}{2}\right)^{-4}$
* Flip the fraction: $\left(\frac{2}{1}\right)^{4}$
* Calculate: $2^4 = 2 \times 2 \times 2 \times 2 = 16$
* Answer: 16

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Row 4

10. $\left(-\frac{2}{8}\right)^{-4}$
* Simplify first: $-\frac{2}{8} = -\frac{1}{4}$
* Problem becomes: $\left(-\frac{1}{4}\right)^{-4}$
* Flip the fraction: $\left(-\frac{4}{1}\right)^{4}$ which is $(-4)^4$
* Calculate: $-4 \times -4 \times -4 \times -4$. Since the exponent is even, the answer is positive. $4^4 = 256$.
* Answer: 256

11. $\left(\frac{8}{16}\right)^{-3}$
* Simplify first: $\frac{8}{16} = \frac{1}{2}$
* Problem becomes: $\left(\frac{1}{2}\right)^{-3}$
* Flip the fraction: $\left(\frac{2}{1}\right)^{3}$
* Calculate: $2^3 = 8$
* Answer: 8

12. $\left(\frac{3}{12}\right)^{-3}$
* Simplify first: $\frac{3}{12} = \frac{1}{4}$
* Problem becomes: $\left(\frac{1}{4}\right)^{-3}$
* Flip the fraction: $\left(\frac{4}{1}\right)^{3}$
* Calculate: $4^3 = 64$
* Answer: 64

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Row 5

13. $\left(\frac{5}{10}\right)^{-5}$
* Simplify first: $\frac{5}{10} = \frac{1}{2}$
* Problem becomes: $\left(\frac{1}{2}\right)^{-5}$
* Flip the fraction: $\left(\frac{2}{1}\right)^{5}$
* Calculate: $2^5 = 32$
* Answer: 32

14. $\left(-\frac{2}{4}\right)^{-5}$
* Simplify first: $-\frac{2}{4} = -\frac{1}{2}$
* Problem becomes: $\left(-\frac{1}{2}\right)^{-5}$
* Flip the fraction: $\left(-\frac{2}{1}\right)^{5}$ which is $(-2)^5$
* Calculate: $-2 \times -2 \times -2 \times -2 \times -2$. Since the exponent is odd, the answer stays negative. $2^5 = 32$, so result is $-32$.
* Answer: -32

15. $\left(\frac{6}{18}\right)^{-2}$
* Simplify first: $\frac{6}{18} = \frac{1}{3}$
* Problem becomes: $\left(\frac{1}{3}\right)^{-2}$
* Flip the fraction: $\left(\frac{3}{1}\right)^{2}$
* Calculate: $3^2 = 9$
* Answer: 9

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Row 6

16. $\left(\frac{4}{16}\right)^{-3}$
* Simplify first: $\frac{4}{16} = \frac{1}{4}$
* Problem becomes: $\left(\frac{1}{4}\right)^{-3}$
* Flip the fraction: $\left(\frac{4}{1}\right)^{3}$
* Calculate: $4^3 = 64$
* Answer: 64

17. $\left(\frac{1}{7}\right)^{-1}$
* Flip the fraction: $\left(\frac{7}{1}\right)^{1}$
* Calculate: $7^1 = 7$
* Answer: 7

18. $\left(-\frac{3}{15}\right)^{-3}$
* Simplify first: $-\frac{3}{15} = -\frac{1}{5}$
* Problem becomes: $\left(-\frac{1}{5}\right)^{-3}$
* Flip the fraction: $\left(-\frac{5}{1}\right)^{3}$ which is $(-5)^3$
* Calculate: $-5 \times -5 \times -5 = -125$
* Answer: -125

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Row 7

19. $\left(\frac{6}{12}\right)^{-8}$
* Simplify first: $\frac{6}{12} = \frac{1}{2}$
* Problem becomes: $\left(\frac{1}{2}\right)^{-8}$
* Flip the fraction: $\left(\frac{2}{1}\right)^{8}$
* Calculate: $2^8 = 256$
* Answer: 256

20. $\left(\frac{1}{8}\right)^{-3}$
* Flip the fraction: $\left(\frac{8}{1}\right)^{3}$
* Calculate: $8^3 = 8 \times 8 \times 8 = 512$
* Answer: 512

21. $\left(\frac{1}{2}\right)^{-1}$
* Flip the fraction: $\left(\frac{2}{1}\right)^{1}$
* Calculate: $2^1 = 2$
* Answer: 2

Final Answer:
1. 25
2. 9
3. 64
4. 49
5. 9
6. 4
7. 216
8. 256
9. 16
10. 256
11. 8
12. 64
13. 32
14. -32
15. 9
16. 64
17. 7
18. -125
19. 256
20. 512
21. 2