Negative Exponents Worksheet for simplifying expressions to positive exponents.

Let’s work through each problem one by one. The rule is: simplify so that only positive exponents appear in the answer.

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Problem 1:
$ 11^{-4} $
→ Negative exponent means “flip to denominator”:
$ \frac{1}{11^4} $

✔ Final: $ \frac{1}{11^4} $

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Problem 2:
$ \frac{1}{6^{-4}} $
→ A negative exponent in the denominator flips up:
$ 6^4 $

✔ Final: $ 6^4 $

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Problem 3:
$ x^{-2} \cdot x^{-4} $
→ When multiplying same base, add exponents:
$ x^{-2 + (-4)} = x^{-6} $
→ Flip to make exponent positive:
$ \frac{1}{x^6} $

✔ Final: $ \frac{1}{x^6} $

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Problem 4:
$ -\frac{x y^{-1}}{9 z^{-2}} $
→ Move terms with negative exponents:
- $ y^{-1} $ goes to denominator → becomes $ y $
- $ z^{-2} $ goes to numerator → becomes $ z^2 $

So:
$ -\frac{x \cdot z^2}{9 \cdot y} $

✔ Final: $ -\frac{x z^2}{9y} $

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Problem 5:
$ \frac{p^{-8}}{p^{-6}} $
→ Subtract exponents (numerator minus denominator):
$ p^{-8 - (-6)} = p^{-8 + 6} = p^{-2} $
→ Flip to make positive:
$ \frac{1}{p^2} $

✔ Final: $ \frac{1}{p^2} $

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Problem 6:
$ 9 q^{-2} $
→ Only $ q $ has negative exponent → move it down:
$ \frac{9}{q^2} $

✔ Final: $ \frac{9}{q^2} $

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Problem 7:
$ \frac{f^{-8}}{f^{-5}} $
→ Subtract exponents:
$ f^{-8 - (-5)} = f^{-8 + 5} = f^{-3} $
→ Flip:
$ \frac{1}{f^3} $

✔ Final: $ \frac{1}{f^3} $

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Problem 8:
$ n^{-12} \cdot n^{-2} $
→ Add exponents:
$ n^{-12 + (-2)} = n^{-14} $
→ Flip:
$ \frac{1}{n^{14}} $

✔ Final: $ \frac{1}{n^{14}} $

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Problem 9:
$ 902^{-1} $
→ Just flip:
$ \frac{1}{902} $

✔ Final: $ \frac{1}{902} $

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Problem 10:
$ 5^{-2} \cdot 5^{-4} \cdot 5^{-1} $
→ Add all exponents:
$ -2 + (-4) + (-1) = -7 $
→ So: $ 5^{-7} $
→ Flip:
$ \frac{1}{5^7} $

✔ Final: $ \frac{1}{5^7} $

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Problem 11:
$ 3 \cdot (3^{-4})^{-2} $
→ First, handle power of a power: multiply exponents inside:
$ (3^{-4})^{-2} = 3^{(-4)(-2)} = 3^8 $
→ Now multiply by 3:
$ 3^1 \cdot 3^8 = 3^{9} $

✔ Final: $ 3^9 $

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Problem 12:
$ (2^{-2})^{-2} \cdot 2^{-1} $
→ First part: $ (2^{-2})^{-2} = 2^{(-2)(-2)} = 2^4 $
→ Then multiply by $ 2^{-1} $:
$ 2^4 \cdot 2^{-1} = 2^{4 + (-1)} = 2^3 $

✔ Final: $ 2^3 $

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Final Answers:

1. $ \frac{1}{11^4} $
2. $ 6^4 $
3. $ \frac{1}{x^6} $
4. $ -\frac{x z^2}{9y} $
5. $ \frac{1}{p^2} $
6. $ \frac{9}{q^2} $
7. $ \frac{1}{f^3} $
8. $ \frac{1}{n^{14}} $
9. $ \frac{1}{902} $
10. $ \frac{1}{5^7} $
11. $ 3^9 $
12. $ 2^3 $

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Final Answer:
1. $\frac{1}{11^4}$
2. $6^4$
3. $\frac{1}{x^6}$
4. $-\frac{x z^2}{9y}$
5. $\frac{1}{p^2}$
6. $\frac{9}{q^2}$
7. $\frac{1}{f^3}$
8. $\frac{1}{n^{14}}$
9. $\frac{1}{902}$
10. $\frac{1}{5^7}$
11. $3^9$
12. $2^3$