Properties of Exponents Practice | Worksheet - Free Printable

Problem: Simplify each expression using the properties of exponents. Write the answer as a single term with a positive exponent.

#### Basic Properties of Exponents Used:
1. Product Rule: $ a^m \cdot a^n = a^{m+n} $
2. Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
3. Power Rule: $ (a^m)^n = a^{m \cdot n} $
4. Negative Exponent Rule: $ a^{-n} = \frac{1}{a^n} $

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Solutions to Each Expression

#### Part 1: Simplify Each Expression

1. $ 4^3 \cdot 4^5 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 4^3 \cdot 4^5 = 4^{3+5} = 4^8 $

2. $ 3^7 \cdot 3^3 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 3^7 \cdot 3^3 = 3^{7+3} = 3^{10} $

3. $ \frac{6^4}{6^9} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{6^4}{6^9} = 6^{4-9} = 6^{-5} $
- Convert to a positive exponent using the Negative Exponent Rule: $ 6^{-5} = \frac{1}{6^5} $

4. $ (5^3)^4 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (5^3)^4 = 5^{3 \cdot 4} = 5^{12} $

5. $ 2^8 \cdot 2^4 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 2^8 \cdot 2^4 = 2^{8+4} = 2^{12} $

6. $ \frac{12^8}{12^3} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{12^8}{12^3} = 12^{8-3} = 12^5 $

7. $ 8^5 \cdot 8^3 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 8^5 \cdot 8^3 = 8^{5+3} = 8^8 $

8. $ (17^3)^4 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (17^3)^4 = 17^{3 \cdot 4} = 17^{12} $

9. $ (13^3)^5 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (13^3)^5 = 13^{3 \cdot 5} = 13^{15} $

10. $ 3^8 \cdot 3^5 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 3^8 \cdot 3^5 = 3^{8+5} = 3^{13} $

11. $ 14^{16} \cdot 14^0 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- Note that $ 14^0 = 1 $, so:
- $ 14^{16} \cdot 14^0 = 14^{16+0} = 14^{16} $

12. $ \frac{9^8}{9^5} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{9^8}{9^5} = 9^{8-5} = 9^3 $

13. $ 7^5 \cdot 7^2 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 7^5 \cdot 7^2 = 7^{5+2} = 7^7 $

14. $ (4^2)^5 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (4^2)^5 = 4^{2 \cdot 5} = 4^{10} $

15. $ \frac{10^{12}}{10^3} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{10^{12}}{10^3} = 10^{12-3} = 10^9 $

16. $ 45^8 \cdot 45^0 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- Note that $ 45^0 = 1 $, so:
- $ 45^8 \cdot 45^0 = 45^{8+0} = 45^8 $

17. $ \frac{2^8}{2^5} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{2^8}{2^5} = 2^{8-5} = 2^3 $

18. $ \frac{18^{10}}{18^7} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{18^{10}}{18^7} = 18^{10-7} = 18^3 $

19. $ (6^3)^4 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (6^3)^4 = 6^{3 \cdot 4} = 6^{12} $

20. $ 5^8 \cdot 5^m $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 5^8 \cdot 5^m = 5^{8+m} $

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#### Part 2: Challenge Problems

1. $ \frac{9^7}{9^4} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{9^7}{9^4} = 9^{7-4} = 9^3 $

2. $ \frac{25^3}{25^2} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- $ \frac{25^3}{25^2} = 25^{3-2} = 25^1 = 25 $

3. $ (7^3)^4 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (7^3)^4 = 7^{3 \cdot 4} = 7^{12} $

4. $ (x^3)^4 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- $ (x^3)^4 = x^{3 \cdot 4} = x^{12} $

5. $ 5^4 \cdot (5^3)^2 $
- First, simplify $ (5^3)^2 $ using the Power Rule:
- $ (5^3)^2 = 5^{3 \cdot 2} = 5^6 $
- Now use the Product Rule:
- $ 5^4 \cdot 5^6 = 5^{4+6} = 5^{10} $

6. $ 6^3 \cdot 6^4 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- $ 6^3 \cdot 6^4 = 6^{3+4} = 6^7 $

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Final Answers

#### Part 1:
1. $ 4^8 $
2. $ 3^{10} $
3. $ \frac{1}{6^5} $
4. $ 5^{12} $
5. $ 2^{12} $
6. $ 12^5 $
7. $ 8^8 $
8. $ 17^{12} $
9. $ 13^{15} $
10. $ 3^{13} $
11. $ 14^{16} $
12. $ 9^3 $
13. $ 7^7 $
14. $ 4^{10} $
15. $ 10^9 $
16. $ 45^8 $
17. $ 2^3 $
18. $ 18^3 $
19. $ 6^{12} $
20. $ 5^{8+m} $

#### Part 2:
1. $ 9^3 $
2. $ 25 $
3. $ 7^{12} $
4. $ x^{12} $
5. $ 5^{10} $
6. $ 6^7 $

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Boxed Final Answer

$$
\boxed{
\begin{aligned}
&\text{Part 1: } 4^8, 3^{10}, \frac{1}{6^5}, 5^{12}, 2^{12}, 12^5, 8^8, 17^{12}, 13^{15}, 3^{13}, 14^{16}, 9^3, 7^7, 4^{10}, 10^9, 45^8, 2^3, 18^3, 6^{12}, 5^{8+m} \\
&\text{Part 2: } 9^3, 25, 7^{12}, x^{12}, 5^{10}, 6^7
\end{aligned}
}
$$