Properties of Exponents Practice | Worksheet - Free Printable

Let’s solve each problem one by one using the properties of exponents. We’ll use these key rules:

1. Product Rule: When multiplying powers with the same base, add the exponents:
\( a^m \cdot a^n = a^{m+n} \)

2. Quotient Rule: When dividing powers with the same base, subtract the exponents:
\( \frac{a^m}{a^n} = a^{m-n} \)

3. Power of a Power Rule: When raising a power to another power, multiply the exponents:
\( (a^m)^n = a^{m \cdot n} \)

4. Negative Exponent Rule: A negative exponent means take the reciprocal and make the exponent positive:
\( a^{-n} = \frac{1}{a^n} \), and \( \frac{1}{a^{-n}} = a^n \)

We will simplify each expression to have a positive exponent.

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First Row:

1. \( 4^5 \cdot 4^7 \)
→ Same base, so add exponents: \( 4^{5+7} = 4^{12} \)

2. \( 3^0 = \)
→ Any number to the 0 power is 1 → \( 1 \)

3. \( \frac{6^{10}}{6^7} \)
→ Subtract exponents: \( 6^{10-7} = 6^3 \)

4. \( (5^3)^4 \)
→ Multiply exponents: \( 5^{3 \cdot 4} = 5^{12} \)

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Second Row:

5. \( 2^4 = \)
→ Just calculate: \( 2 \cdot 2 \cdot 2 \cdot 2 = 16 \)

6. \( \frac{12^8}{12^4} \)
→ Subtract exponents: \( 12^{8-4} = 12^4 \)

7. \( 8^1 \cdot 8^5 \)
→ Add exponents: \( 8^{1+5} = 8^6 \)

8. \( (17^0)^9 \)
→ First, \( 17^0 = 1 \), then \( 1^9 = 1 \)

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Third Row:

9. \( (13^2)^9 \)
→ Multiply exponents: \( 13^{2 \cdot 9} = 13^{18} \)

10. \( 3^7 \cdot 3^8 \)
→ Add exponents: \( 3^{7+8} = 3^{15} \)

11. \( 14^{-6} = \)
→ Negative exponent → flip: \( \frac{1}{14^6} \)

12. \( \frac{3^{21}}{9^6} \)
→ Note: 9 = 3², so rewrite denominator: \( 9^6 = (3^2)^6 = 3^{12} \)
Now: \( \frac{3^{21}}{3^{12}} = 3^{21-12} = 3^9 \)

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Fourth Row:

13. \( 7^5 \cdot 7^9 \)
→ Add exponents: \( 7^{5+9} = 7^{14} \)

14. \( (4^2)^5 \)
→ Multiply exponents: \( 4^{2 \cdot 5} = 4^{10} \)

15. \( \frac{10^{10}}{10^{18}} \)
→ Subtract exponents: \( 10^{10-18} = 10^{-8} \) → flip for positive: \( \frac{1}{10^8} \)

16. \( 48^0 = \)
→ Anything to the 0 power is 1 → \( 1 \)

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Fifth Row:

17. \( \frac{2^{10}}{2^5} \)
→ Subtract exponents: \( 2^{10-5} = 2^5 \)

18. \( 18^{-5} = \)
→ Flip for positive exponent: \( \frac{1}{18^5} \)

19. \( (6^{-3})^2 \)
→ Multiply exponents: \( 6^{-6} \) → flip: \( \frac{1}{6^6} \)

20. \( 5^4 \cdot 5^{-6} \)
→ Add exponents: \( 5^{4 + (-6)} = 5^{-2} \) → flip: \( \frac{1}{5^2} \)

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Challenge Section:

21. \( \frac{9^5}{9^8} \)
→ Subtract exponents: \( 9^{5-8} = 9^{-3} \) → flip: \( \frac{1}{9^3} \)

22. \( \frac{25^7}{25^2} \cdot 25^0 \)
→ First, \( 25^0 = 1 \), so ignore it.
Then: \( \frac{25^7}{25^2} = 25^{7-2} = 25^5 \)

23. \( \frac{(7^3)^4}{7^5} \)
→ Numerator: \( (7^3)^4 = 7^{12} \)
Then: \( \frac{7^{12}}{7^5} = 7^{12-5} = 7^7 \)

24. \( (4^{-3})^5 \)
→ Multiply exponents: \( 4^{-15} \) → flip: \( \frac{1}{4^{15}} \)

25. \( 5^4 \cdot (5^3)^{-2} \)
→ First, \( (5^3)^{-2} = 5^{-6} \)
Then: \( 5^4 \cdot 5^{-6} = 5^{4-6} = 5^{-2} \) → flip: \( \frac{1}{5^2} \)

26. \( \frac{6^{-4} \cdot 6^5}{6^4} \)
→ Numerator: \( 6^{-4+5} = 6^1 \)
Then: \( \frac{6^1}{6^4} = 6^{1-4} = 6^{-3} \) → flip: \( \frac{1}{6^3} \)

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Final Answer:

First Row:
1. \( 4^{12} \)
2. \( 1 \)
3. \( 6^3 \)
4. \( 5^{12} \)

Second Row:
5. \( 16 \)
6. \( 12^4 \)
7. \( 8^6 \)
8. \( 1 \)

Third Row:
9. \( 13^{18} \)
10. \( 3^{15} \)
11. \( \frac{1}{14^6} \)
12. \( 3^9 \)

Fourth Row:
13. \( 7^{14} \)
14. \( 4^{10} \)
15. \( \frac{1}{10^8} \)
16. \( 1 \)

Fifth Row:
17. \( 2^5 \)
18. \( \frac{1}{18^5} \)
19. \( \frac{1}{6^6} \)
20. \( \frac{1}{5^2} \)

Challenge:
21. \( \frac{1}{9^3} \)
22. \( 25^5 \)
23. \( 7^7 \)
24. \( \frac{1}{4^{15}} \)
25. \( \frac{1}{5^2} \)
26. \( \frac{1}{6^3} \)