Negative Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem on the Negative and Zero Exponents Worksheet step by step. Remember the key rules:

Exponent Rules:

1. $ a^{-n} = \frac{1}{a^n} $
2. $ a^0 = 1 $ (for $ a \neq 0 $)
3. $ (ab)^n = a^n b^n $
4. $ \frac{a^m}{a^n} = a^{m-n} $
5. $ a^m \cdot a^n = a^{m+n} $

---

1. $ 6^{-3} $

Use rule: $ a^{-n} = \frac{1}{a^n} $

$$
6^{-3} = \frac{1}{6^3} = \frac{1}{216}
$$

✔ Answer: $ \frac{1}{216} $

---

2. $ 4^0 $

Any nonzero number to the power of 0 is 1.

$$
4^0 = 1
$$

✔ Answer: $ 1 $

---

3. $ (3pq^0)^{-2} $

First, simplify inside the parentheses:

- $ q^0 = 1 $
- So $ 3pq^0 = 3p(1) = 3p $

Now:
$$
(3p)^{-2} = \frac{1}{(3p)^2} = \frac{1}{9p^2}
$$

✔ Answer: $ \frac{1}{9p^2} $

---

4. $ (5p^4)^0 $

Any nonzero expression raised to the 0 power is 1.

$$
(5p^4)^0 = 1
$$

✔ Answer: $ 1 $

---

5. $ -6x^{-1}y^3 $

Move $ x^{-1} $ to the denominator:

$$
-6x^{-1}y^3 = -\frac{6y^3}{x}
$$

✔ Answer: $ -\frac{6y^3}{x} $

---

6. $ -p^4q^{-1} $

Move $ q^{-1} $ to the denominator:

$$
-p^4q^{-1} = -\frac{p^4}{q}
$$

✔ Answer: $ -\frac{p^4}{q} $

---

7. $ \frac{3m^{-5}n^{-6}}{9m^8n^0} $

Step-by-step:

- Simplify constants: $ \frac{3}{9} = \frac{1}{3} $
- $ n^0 = 1 $
- Combine exponents:

$$
m^{-5 - 8} = m^{-13}, \quad n^{-6 - 0} = n^{-6}
$$

So:
$$
\frac{1}{3} m^{-13} n^{-6} = \frac{1}{3m^{13}n^6}
$$

✔ Answer: $ \frac{1}{3m^{13}n^6} $

---

8. $ 6x^{-2} \cdot 6x^0 $

Multiply coefficients: $ 6 \cdot 6 = 36 $

Exponents:
- $ x^{-2} \cdot x^0 = x^{-2 + 0} = x^{-2} $

So:
$$
36x^{-2} = \frac{36}{x^2}
$$

✔ Answer: $ \frac{36}{x^2} $

---

9. $ 3x^{-3}y^0 $

- $ y^0 = 1 $
- $ x^{-3} = \frac{1}{x^3} $

So:
$$
3x^{-3}y^0 = 3 \cdot \frac{1}{x^3} \cdot 1 = \frac{3}{x^3}
$$

✔ Answer: $ \frac{3}{x^3} $

---

10. $ 2u^0v^{-3} $

- $ u^0 = 1 $
- $ v^{-3} = \frac{1}{v^3} $

So:
$$
2 \cdot 1 \cdot \frac{1}{v^3} = \frac{2}{v^3}
$$

✔ Answer: $ \frac{2}{v^3} $

---

11. $ \left(\frac{5}{3}\right)^{-3} $

Apply negative exponent rule:

$$
\left(\frac{5}{3}\right)^{-3} = \left(\frac{3}{5}\right)^3 = \frac{3^3}{5^3} = \frac{27}{125}
$$

✔ Answer: $ \frac{27}{125} $

---

12. $ \frac{x}{3^{-3}} $

First, simplify $ 3^{-3} = \frac{1}{3^3} = \frac{1}{27} $

So:
$$
\frac{x}{\frac{1}{27}} = x \cdot 27 = 27x
$$

✔ Answer: $ 27x $

---

✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{1}{216} $ |
| 2 | $ 1 $ |
| 3 | $ \frac{1}{9p^2} $ |
| 4 | $ 1 $ |
| 5 | $ -\frac{6y^3}{x} $ |
| 6 | $ -\frac{p^4}{q} $ |
| 7 | $ \frac{1}{3m^{13}n^6} $ |
| 8 | $ \frac{36}{x^2} $ |
| 9 | $ \frac{3}{x^3} $ |
| 10 | $ \frac{2}{v^3} $ |
| 11 | $ \frac{27}{125} $ |
| 12 | $ 27x $ |

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