Zero and Negative Exponents worksheets - Free Printable

Let's solve each of these expressions step by step using the rules of zero and negative exponents.

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🔷 Rules to Remember:

1. Negative Exponent Rule:
$$
a^{-n} = \frac{1}{a^n}
$$

2. Zero Exponent Rule:
$$
a^0 = 1 \quad \text{(for } a \neq 0\text{)}
$$

3. Zero Base with Positive Exponent:
$$
0^n = 0 \quad \text{(for } n > 0\text{)}
$$

4. Zero Base with Zero Exponent:
$$
0^0 \text{ is undefined}
$$

5. Negative Sign in Front:
If there's a minus sign in front, like $-a^{-n}$, it means:
$$
-a^{-n} = -\left(\frac{1}{a^n}\right)
$$
But if it's $(-a)^{-n}$, then it's $\frac{1}{(-a)^n}$ — but here, the expression is written as $-a^{-n}$, so the negative is not part of the base.

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Now let’s evaluate each:

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✔ Solutions:

1) $ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} $

2) $ 10^{-6} = \frac{1}{10^6} = \frac{1}{1,000,000} $

3) $ 0^8 = 0 $ (since $0^n = 0$ for $n > 0$)

4) $ -10^{-6} = -\left(\frac{1}{10^6}\right) = -\frac{1}{1,000,000} $

5) $ -9^{-3} = -\left(\frac{1}{9^3}\right) = -\frac{1}{729} $

6) $ -11^{-2} = -\left(\frac{1}{11^2}\right) = -\frac{1}{121} $

7) $ -12^{-4} = -\left(\frac{1}{12^4}\right) = -\frac{1}{20736} $

8) $ -3^{-5} = -\left(\frac{1}{3^5}\right) = -\frac{1}{243} $

9) $ 9^{-4} = \frac{1}{9^4} = \frac{1}{6561} $

10) $ 0^2 = 0 $

11) $ -8^{-2} = -\left(\frac{1}{8^2}\right) = -\frac{1}{64} $

12) $ 6^{-5} = \frac{1}{6^5} = \frac{1}{7776} $

13) $ -7^{-2} = -\left(\frac{1}{7^2}\right) = -\frac{1}{49} $

14) $ 0^5 = 0 $

15) $ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $

16) $ 0^6 = 0 $

17) $ 12^{-4} = \frac{1}{12^4} = \frac{1}{20736} $

18) $ 5^{-3} = \frac{1}{5^3} = \frac{1}{125} $

19) $ -4^{-3} = -\left(\frac{1}{4^3}\right) = -\frac{1}{64} $

20) $ -5^{-3} = -\left(\frac{1}{5^3}\right) = -\frac{1}{125} $

21) $ -6^{-4} = -\left(\frac{1}{6^4}\right) = -\frac{1}{1296} $

22) $ 0^7 = 0 $

23) $ 10^{-6} = \frac{1}{10^6} = \frac{1}{1,000,000} $

24) $ 10^{-2} = \frac{1}{10^2} = \frac{1}{100} $

25) $ -2^{-5} = -\left(\frac{1}{2^5}\right) = -\frac{1}{32} $

26) $ 8^{-2} = \frac{1}{8^2} = \frac{1}{64} $

27) $ 7^{-3} = \frac{1}{7^3} = \frac{1}{343} $

28) $ 10^{-4} = \frac{1}{10^4} = \frac{1}{10,000} $

29) $ 4^{-5} = \frac{1}{4^5} = \frac{1}{1024} $

30) $ 0^4 = 0 $

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📌 Final Answers:

| Problem | Answer |
|--------|--------|
| 1) $3^{-4}$ | $\frac{1}{81}$ |
| 2) $10^{-6}$ | $\frac{1}{1,000,000}$ |
| 3) $0^8$ | $0$ |
| 4) $-10^{-6}$ | $-\frac{1}{1,000,000}$ |
| 5) $-9^{-3}$ | $-\frac{1}{729}$ |
| 6) $-11^{-2}$ | $-\frac{1}{121}$ |
| 7) $-12^{-4}$ | $-\frac{1}{20736}$ |
| 8) $-3^{-5}$ | $-\frac{1}{243}$ |
| 9) $9^{-4}$ | $\frac{1}{6561}$ |
| 10) $0^2$ | $0$ |
| 11) $-8^{-2}$ | $-\frac{1}{64}$ |
| 12) $6^{-5}$ | $\frac{1}{7776}$ |
| 13) $-7^{-2}$ | $-\frac{1}{49}$ |
| 14) $0^5$ | $0$ |
| 15) $2^{-3}$ | $\frac{1}{8}$ |
| 16) $0^6$ | $0$ |
| 17) $12^{-4}$ | $\frac{1}{20736}$ |
| 18) $5^{-3}$ | $\frac{1}{125}$ |
| 19) $-4^{-3}$ | $-\frac{1}{64}$ |
| 20) $-5^{-3}$ | $-\frac{1}{125}$ |
| 21) $-6^{-4}$ | $-\frac{1}{1296}$ |
| 22) $0^7$ | $0$ |
| 23) $10^{-6}$ | $\frac{1}{1,000,000}$ |
| 24) $10^{-2}$ | $\frac{1}{100}$ |
| 25) $-2^{-5}$ | $-\frac{1}{32}$ |
| 26) $8^{-2}$ | $\frac{1}{64}$ |
| 27) $7^{-3}$ | $\frac{1}{343}$ |
| 28) $10^{-4}$ | $\frac{1}{10,000}$ |
| 29) $4^{-5}$ | $\frac{1}{1024}$ |
| 30) $0^4$ | $0$ |

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✔ All answers are now evaluated based on exponent rules. Let me know if you'd like them in decimal form or simplified fractions!