Negative Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem on the Negative Exponents Worksheet step by step, applying the rules of exponents. The key rule for negative exponents is:

> $ a^{-n} = \frac{1}{a^n} $

Also recall:
- $ a^m \cdot a^n = a^{m+n} $
- $ \frac{a^m}{a^n} = a^{m-n} $
- $ (a^m)^n = a^{m \cdot n} $
- $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $

We will simplify each expression so that only positive exponents remain.

---

1. $ 11^{-4} $

$$
11^{-4} = \frac{1}{11^4}
$$

✔ Final Answer: $ \boxed{\frac{1}{11^4}} $

---

2. $ \frac{1}{6^{-4}} $

$$
\frac{1}{6^{-4}} = 6^4 \quad \text{(because } \frac{1}{a^{-n}} = a^n \text{)}
$$

✔ Final Answer: $ \boxed{6^4} $

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3. $ x^{-2} \cdot x^{-4} $

Use the product rule: $ x^{-2} \cdot x^{-4} = x^{-2 + (-4)} = x^{-6} $

Now convert to positive exponent:

$$
x^{-6} = \frac{1}{x^6}
$$

✔ Final Answer: $ \boxed{\frac{1}{x^6}} $

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4. $ \frac{xy^{-1}}{9z^{-2}} $

Break it down:

- $ y^{-1} = \frac{1}{y} $
- $ z^{-2} = \frac{1}{z^2} $, so $ \frac{1}{z^{-2}} = z^2 $

So rewrite:

$$
\frac{x \cdot \frac{1}{y}}{9 \cdot \frac{1}{z^2}} = \frac{x}{y} \cdot \frac{z^2}{9} = \frac{x z^2}{9y}
$$

✔ Final Answer: $ \boxed{\frac{x z^2}{9y}} $

---

5. $ \frac{p^8}{p^6} $

$$
\frac{p^8}{p^6} = p^{8-6} = p^2
$$

✔ Final Answer: $ \boxed{p^2} $

---

6. $ 9q^{-2} $

$$
9q^{-2} = 9 \cdot \frac{1}{q^2} = \frac{9}{q^2}
$$

✔ Final Answer: $ \boxed{\frac{9}{q^2}} $

---

7. $ \frac{f^{-8}}{f^{-5}} $

$$
\frac{f^{-8}}{f^{-5}} = f^{-8 - (-5)} = f^{-3} = \frac{1}{f^3}
$$

✔ Final Answer: $ \boxed{\frac{1}{f^3}} $

---

8. $ n^{-12} \cdot n^{-2} $

$$
n^{-12} \cdot n^{-2} = n^{-14} = \frac{1}{n^{14}}
$$

✔ Final Answer: $ \boxed{\frac{1}{n^{14}}} $

---

9. $ 902^{-1} $

$$
902^{-1} = \frac{1}{902}
$$

✔ Final Answer: $ \boxed{\frac{1}{902}} $

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10. $ 5^{-2} \cdot 5^{-4} \cdot 5^{-1} $

Add exponents:

$$
5^{-2 + (-4) + (-1)} = 5^{-7} = \frac{1}{5^7}
$$

✔ Final Answer: $ \boxed{\frac{1}{5^7}} $

---

11. $ 3 \cdot (3^4)^{-2} $

First simplify $ (3^4)^{-2} $:

$$
(3^4)^{-2} = 3^{4 \cdot (-2)} = 3^{-8}
$$

Now multiply:

$$
3 \cdot 3^{-8} = 3^{1 + (-8)} = 3^{-7} = \frac{1}{3^7}
$$

✔ Final Answer: $ \boxed{\frac{1}{3^7}} $

---

12. $ (2^{-2})^{-2} \cdot 2^{-1} $

First simplify $ (2^{-2})^{-2} $:

$$
(2^{-2})^{-2} = 2^{(-2) \cdot (-2)} = 2^4
$$

Now multiply by $ 2^{-1} $:

$$
2^4 \cdot 2^{-1} = 2^{4 - 1} = 2^3 = 8
$$

✔ Final Answer: $ \boxed{8} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{1}{11^4} $ |
| 2 | $ 6^4 $ |
| 3 | $ \frac{1}{x^6} $ |
| 4 | $ \frac{x z^2}{9y} $ |
| 5 | $ 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 | $ \frac{1}{3^7} $ |
| 12 | $ 8 $ |

All answers contain only positive exponents, as required.

Let me know if you'd like this in printable format or with explanations for any specific step!