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

Let’s go through each problem one by one and check if the equation is true or false.

We’ll use basic exponent rules:

- When multiplying same base: \( a^m \cdot a^n = a^{m+n} \)
- When dividing same base: \( \frac{a^m}{a^n} = a^{m-n} \)
- Power of a power: \( (a^m)^n = a^{m \cdot n} \)
- Any number to the power 1 is itself: \( a^1 = a \)
- If bases are different, you can’t combine exponents unless they’re equal after simplifying.

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Problem 1: \( 3^2 \cdot 3^3 = 3^5 \)
→ Same base? Yes. Add exponents: 2 + 3 = 5 → \( 3^5 \). ✔ True

Problem 2: \( (8^3)^4 = 8^7 \)
→ Power of a power: multiply exponents → 3 × 4 = 12 → should be \( 8^{12} \), not \( 8^7 \). ✘ False

Problem 3: \( \frac{4^6}{4^2} = 4^3 \)
→ Divide: subtract exponents → 6 - 2 = 4 → should be \( 4^4 \), not \( 4^3 \). ✘ False

Problem 4: \( \frac{9^8}{9^1} = 9^7 \)
→ Subtract exponents: 8 - 1 = 7 → \( 9^7 \). ✔ True

Problem 5: \( 5^2 \cdot 5^2 = 5^4 \)
→ Add exponents: 2 + 2 = 4 → \( 5^4 \). ✔ True

Problem 6: \( 1 = 2^0 \)
→ Anything (except 0) to the power 0 is 1 → \( 2^0 = 1 \). ✔ True

Problem 7: \( (2^3)^2 = 2^5 \)
→ Multiply exponents: 3 × 2 = 6 → should be \( 2^6 \), not \( 2^5 \). ✘ False

Problem 8: \( 5^1 = \frac{1}{5} \)
→ \( 5^1 = 5 \), but \( \frac{1}{5} = 5^{-1} \). So this is wrong. ✘ False

Problem 9: \( 9^1 \cdot 9^3 = 9^3 \)
→ Add exponents: 1 + 3 = 4 → should be \( 9^4 \), not \( 9^3 \). ✘ False

Problem 10: \( \frac{6^7}{6^1} = 6^6 \)
→ Subtract: 7 - 1 = 6 → \( 6^6 \). ✔ True

Problem 11: \( (7^2) \cdot 7^0 = 7^2 \)
→ \( 7^0 = 1 \), so \( 7^2 \cdot 1 = 7^2 \). ✔ True

Problem 12: \( \frac{2^{12}}{2^3} = 2^9 \)
→ Subtract: 12 - 3 = 9 → \( 2^9 \). ✔ True

Problem 13: \( \frac{(4^3)^2}{4^2} = 4^4 \)
→ First, numerator: \( (4^3)^2 = 4^{6} \)
→ Then divide: \( \frac{4^6}{4^2} = 4^{6-2} = 4^4 \). ✔ True

Problem 14: \( \frac{12^3}{12^1} = \frac{1}{12^2} \)
→ Left side: \( 12^{3-1} = 12^2 \)
→ Right side: \( \frac{1}{12^2} = 12^{-2} \)
→ Not equal. ✘ False

Problem 15: \( \frac{4^2 \cdot 4^6}{4^4} = 4^4 \)
→ Numerator: \( 4^{2+6} = 4^8 \)
→ Divide: \( \frac{4^8}{4^4} = 4^{8-4} = 4^4 \). ✔ True

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Now let’s list all answers clearly:

1. True
2. False
3. False
4. True
5. True
6. True
7. False
8. False
9. False
10. True
11. True
12. True
13. True
14. False
15. True

✔ Final Answer:
1. True
2. False
3. False
4. True
5. True
6. True
7. False
8. False
9. False
10. True
11. True
12. True
13. True
14. False
15. True