Properties of Exponents: True or False | Worksheet - Free Printable

Let's solve each of the "Properties of Exponents: True or False" problems step by step, using exponent rules. Then we’ll address the Challenge at the end.

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🔹 Exponent Rules to Remember:

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

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Now let’s go through each problem:

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1. $ 3^2 \cdot 3^3 = 3^5 $

- Use rule: $ a^m \cdot a^n = a^{m+n} $
- $ 3^2 \cdot 3^3 = 3^{2+3} = 3^5 $
- ✔ True

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2. $ (8^3)^4 = 8^7 $

- Use rule: $ (a^m)^n = a^{m \cdot n} $
- $ (8^3)^4 = 8^{3 \cdot 4} = 8^{12} $
- But right side is $ 8^7 $
- ✘ False

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3. $ \frac{4^5}{4^2} = 4^3 $

- Use rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{4^5}{4^2} = 4^{5-2} = 4^3 $
- ✔ True

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4. $ \frac{9^6}{9^3} = 9^2 $

- $ \frac{9^6}{9^3} = 9^{6-3} = 9^3 $
- Right side is $ 9^2 $
- ✘ False

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5. $ 5^3 \cdot 5^2 = 5^6 $

- $ 5^3 \cdot 5^2 = 5^{3+2} = 5^5 $
- But right side is $ 5^6 $
- ✘ False

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6. $ 1 = 2^0 $

- Any nonzero number to the power of 0 is 1
- $ 2^0 = 1 $
- ✔ True

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7. $ (2^3)^2 = 2^5 $

- $ (2^3)^2 = 2^{3 \cdot 2} = 2^6 $
- Right side is $ 2^5 $
- ✘ False

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8. $ 5^{-1} = \frac{1}{5^1} $

- Negative exponent: $ a^{-n} = \frac{1}{a^n} $
- So $ 5^{-1} = \frac{1}{5} $
- ✔ True

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9. $ 9^2 \cdot 9^3 = 9^5 $

- $ 9^2 \cdot 9^3 = 9^{2+3} = 9^5 $
- ✔ True

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10. $ \frac{6^6}{6^4} = 6^2 $

- $ \frac{6^6}{6^4} = 6^{6-4} = 6^2 $
- ✔ True

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11. $ (7^2)^3 = 7^6 $

- $ (7^2)^3 = 7^{2 \cdot 3} = 7^6 $
- ✔ True

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12. $ \frac{1}{2^{10}} = 2^{-10} $

- $ \frac{1}{a^n} = a^{-n} $
- So $ \frac{1}{2^{10}} = 2^{-10} $
- ✔ True

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13. $ \frac{(4^2)^3}{4^2} = 4^4 $

- First simplify numerator: $ (4^2)^3 = 4^{2 \cdot 3} = 4^6 $
- Now divide: $ \frac{4^6}{4^2} = 4^{6-2} = 4^4 $
- ✔ True

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14. $ \frac{12^5}{12^3} = \frac{1}{12^2} $

- Left side: $ \frac{12^5}{12^3} = 12^{5-3} = 12^2 $
- Right side: $ \frac{1}{12^2} $
- $ 12^2 \neq \frac{1}{12^2} $
- ✘ False

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15. $ \frac{4^7 \cdot 4^3}{4^6} = 4^{10} $

- Numerator: $ 4^7 \cdot 4^3 = 4^{10} $
- Then: $ \frac{4^{10}}{4^6} = 4^{10-6} = 4^4 $
- But right side says $ 4^{10} $
- ✘ False

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## ✔ Final Answers:

| # | Statement | True / False |
|---|-----------|--------------|
| 1 | $ 3^2 \cdot 3^3 = 3^5 $ | True |
| 2 | $ (8^3)^4 = 8^7 $ | False |
| 3 | $ \frac{4^5}{4^2} = 4^3 $ | True |
| 4 | $ \frac{9^6}{9^3} = 9^2 $ | False |
| 5 | $ 5^3 \cdot 5^2 = 5^6 $ | False |
| 6 | $ 1 = 2^0 $ | True |
| 7 | $ (2^3)^2 = 2^5 $ | False |
| 8 | $ 5^{-1} = \frac{1}{5^1} $ | True |
| 9 | $ 9^2 \cdot 9^3 = 9^5 $ | True |
|10 | $ \frac{6^6}{6^4} = 6^2 $ | True |
|11 | $ (7^2)^3 = 7^6 $ | True |
|12 | $ \frac{1}{2^{10}} = 2^{-10} $ | True |
|13 | $ \frac{(4^2)^3}{4^2} = 4^4 $ | True |
|14 | $ \frac{12^5}{12^3} = \frac{1}{12^2} $ | False |
|15 | $ \frac{4^7 \cdot 4^3}{4^6} = 4^{10} $ | False |

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## 🚩 CHALLENGE:
> Correct all the false equations. Change one side of each false equation to make it true.

We'll fix only the false ones: #2, #4, #5, #7, #14, #15

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2. $ (8^3)^4 = 8^7 $ → False

- Correct version: $ (8^3)^4 = 8^{12} $
- Fix: Change $ 8^7 $ → $ 8^{12} $
- ✔ $ (8^3)^4 = 8^{12} $

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4. $ \frac{9^6}{9^3} = 9^2 $ → False

- $ \frac{9^6}{9^3} = 9^3 $
- Fix: Change $ 9^2 $ → $ 9^3 $
- ✔ $ \frac{9^6}{9^3} = 9^3 $

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5. $ 5^3 \cdot 5^2 = 5^6 $ → False

- $ 5^3 \cdot 5^2 = 5^5 $
- Fix: Change $ 5^6 $ → $ 5^5 $
- ✔ $ 5^3 \cdot 5^2 = 5^5 $

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7. $ (2^3)^2 = 2^5 $ → False

- $ (2^3)^2 = 2^6 $
- Fix: Change $ 2^5 $ → $ 2^6 $
- ✔ $ (2^3)^2 = 2^6 $

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14. $ \frac{12^5}{12^3} = \frac{1}{12^2} $ → False

- $ \frac{12^5}{12^3} = 12^2 $
- Fix: Change $ \frac{1}{12^2} $ → $ 12^2 $
- ✔ $ \frac{12^5}{12^3} = 12^2 $

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15. $ \frac{4^7 \cdot 4^3}{4^6} = 4^{10} $ → False

- We found earlier: LHS = $ 4^4 $
- Fix: Change $ 4^{10} $ → $ 4^4 $
- ✔ $ \frac{4^7 \cdot 4^3}{4^6} = 4^4 $

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## ✔ Challenge Answers:

| Original Equation | Corrected Version |
|-------------------|-------------------|
| $ (8^3)^4 = 8^7 $ | $ (8^3)^4 = 8^{12} $ |
| $ \frac{9^6}{9^3} = 9^2 $ | $ \frac{9^6}{9^3} = 9^3 $ |
| $ 5^3 \cdot 5^2 = 5^6 $ | $ 5^3 \cdot 5^2 = 5^5 $ |
| $ (2^3)^2 = 2^5 $ | $ (2^3)^2 = 2^6 $ |
| $ \frac{12^5}{12^3} = \frac{1}{12^2} $ | $ \frac{12^5}{12^3} = 12^2 $ |
| $ \frac{4^7 \cdot 4^3}{4^6} = 4^{10} $ | $ \frac{4^7 \cdot 4^3}{4^6} = 4^4 $ |

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