Properties of Exponents Practice | Worksheet - Free Printable

Let's solve each problem in the "Properties of Exponents Practice" worksheet step by step. We'll use the laws of exponents to simplify each expression and write the answer as a single term with a positive exponent.

---

🔷 Basic Properties of Exponents:

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

---

## ✔ Part 1: Simplify Each Expression

1. $ 4^2 \cdot 4^3 $

Use: $ a^m \cdot a^n = a^{m+n} $

$$
4^{2+3} = 4^5
$$

✔ Answer: $ 4^5 $

---

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

$$
3^{2+5} = 3^7
$$

✔ Answer: $ 3^7 $

---

3. $ \frac{6^8}{6^3} $

$$
6^{8-3} = 6^5
$$

✔ Answer: $ 6^5 $

---

4. $ (5^3)^4 $

Use: $ (a^m)^n = a^{m \cdot n} $

$$
5^{3 \cdot 4} = 5^{12}
$$

✔ Answer: $ 5^{12} $

---

5. $ 7^4 \cdot 7^6 $

$$
7^{4+6} = 7^{10}
$$

✔ Answer: $ 7^{10} $

---

6. $ \frac{12^4}{12^2} $

$$
12^{4-2} = 12^2
$$

✔ Answer: $ 12^2 $

---

7. $ 8^3 \cdot 8^2 $

$$
8^{3+2} = 8^5
$$

✔ Answer: $ 8^5 $

---

8. $ (17^3)^4 $

$$
17^{3 \cdot 4} = 17^{12}
$$

✔ Answer: $ 17^{12} $

---

9. $ (13^2)^3 $

$$
13^{2 \cdot 3} = 13^6
$$

✔ Answer: $ 13^6 $

---

10. $ 3^2 \cdot 3^4 $

$$
3^{2+4} = 3^6
$$

✔ Answer: $ 3^6 $

---

11. $ 14^4 \div 14^2 $

$$
14^{4-2} = 14^2
$$

✔ Answer: $ 14^2 $

---

12. $ \frac{3^6}{3^3} $

$$
3^{6-3} = 3^3
$$

✔ Answer: $ 3^3 $

---

13. $ 7^2 \cdot 7^6 $

$$
7^{2+6} = 7^8
$$

✔ Answer: $ 7^8 $

---

14. $ (4^3)^2 $

$$
4^{3 \cdot 2} = 4^6
$$

✔ Answer: $ 4^6 $

---

15. $ \frac{10^6}{10^4} $

$$
10^{6-4} = 10^2
$$

✔ Answer: $ 10^2 $

---

16. $ 48^5 \div 48^3 $

$$
48^{5-3} = 48^2
$$

✔ Answer: $ 48^2 $

---

17. $ \frac{2x^5}{x^2} $

Break it down:

$$
\frac{2}{1} \cdot \frac{x^5}{x^2} = 2x^{5-2} = 2x^3
$$

✔ Answer: $ 2x^3 $

---

18. $ 18^5 \div 18^2 $

$$
18^{5-2} = 18^3
$$

✔ Answer: $ 18^3 $

---

19. $ (6^3)^2 $

$$
6^{3 \cdot 2} = 6^6
$$

✔ Answer: $ 6^6 $

---

20. $ 5^{-2} \cdot 5^3 $

Use: $ a^m \cdot a^n = a^{m+n} $

$$
5^{-2+3} = 5^1 = 5
$$

✔ Answer: $ 5 $

---

## ✔ Challenge Problems

Now let’s do the challenge section. These require more steps.

---

21. $ \frac{8^4}{8^3} $

$$
8^{4-3} = 8^1 = 8
$$

✔ Answer: $ 8 $

---

22. $ 25^2 \cdot 25^3 $

$$
25^{2+3} = 25^5
$$

✔ Answer: $ 25^5 $

---

23. $ \frac{(7^3)^2}{7^4} $

First, simplify numerator:

$$
(7^3)^2 = 7^{6}
$$

Then divide:

$$
\frac{7^6}{7^4} = 7^{6-4} = 7^2
$$

✔ Answer: $ 7^2 $

---

24. $ (4^2)^3 $

$$
4^{2 \cdot 3} = 4^6
$$

✔ Answer: $ 4^6 $

---

25. $ 5^2 \cdot (5^3)^2 $

First, simplify $ (5^3)^2 = 5^{6} $

Then:

$$
5^2 \cdot 5^6 = 5^{2+6} = 5^8
$$

✔ Answer: $ 5^8 $

---

26. $ \frac{6^7 \cdot 6^3}{6^4} $

First, multiply numerator:

$$
6^7 \cdot 6^3 = 6^{10}
$$

Then divide:

$$
\frac{6^{10}}{6^4} = 6^{10-4} = 6^6
$$

✔ Answer: $ 6^6 $

---

## ✔ Final Answers Summary

| Problem | Answer |
|--------|--------|
| 1. $ 4^2 \cdot 4^3 $ | $ 4^5 $ |
| 2. $ 3^2 \cdot 3^5 $ | $ 3^7 $ |
| 3. $ \frac{6^8}{6^3} $ | $ 6^5 $ |
| 4. $ (5^3)^4 $ | $ 5^{12} $ |
| 5. $ 7^4 \cdot 7^6 $ | $ 7^{10} $ |
| 6. $ \frac{12^4}{12^2} $ | $ 12^2 $ |
| 7. $ 8^3 \cdot 8^2 $ | $ 8^5 $ |
| 8. $ (17^3)^4 $ | $ 17^{12} $ |
| 9. $ (13^2)^3 $ | $ 13^6 $ |
| 10. $ 3^2 \cdot 3^4 $ | $ 3^6 $ |
| 11. $ 14^4 \div 14^2 $ | $ 14^2 $ |
| 12. $ \frac{3^6}{3^3} $ | $ 3^3 $ |
| 13. $ 7^2 \cdot 7^6 $ | $ 7^8 $ |
| 14. $ (4^3)^2 $ | $ 4^6 $ |
| 15. $ \frac{10^6}{10^4} $ | $ 10^2 $ |
| 16. $ 48^5 \div 48^3 $ | $ 48^2 $ |
| 17. $ \frac{2x^5}{x^2} $ | $ 2x^3 $ |
| 18. $ 18^5 \div 18^2 $ | $ 18^3 $ |
| 19. $ (6^3)^2 $ | $ 6^6 $ |
| 20. $ 5^{-2} \cdot 5^3 $ | $ 5 $ |
| Challenge | |
| 21. $ \frac{8^4}{8^3} $ | $ 8 $ |
| 22. $ 25^2 \cdot 25^3 $ | $ 25^5 $ |
| 23. $ \frac{(7^3)^2}{7^4} $ | $ 7^2 $ |
| 24. $ (4^2)^3 $ | $ 4^6 $ |
| 25. $ 5^2 \cdot (5^3)^2 $ | $ 5^8 $ |
| 26. $ \frac{6^7 \cdot 6^3}{6^4} $ | $ 6^6 $ |

---

✔ Tips for Remembering:

- When multiplying same bases → add exponents
- When dividing same bases → subtract exponents
- Power to a power → multiply exponents
- Negative exponents → move to denominator (or vice versa)
- Always aim for positive exponents in final answer.

Let me know if you'd like this printed or formatted as a PDF!