Independent practice worksheet for simplifying integer exponents, featuring a playful bat illustration.

Let's solve each problem on the "Properties of Integer Exponents – Independent Practice Worksheet" step by step. The worksheet involves simplifying expressions using the laws of exponents.

---

Problem 1: Simplify $6^2 \times 6^4$

Rule: When multiplying powers with the same base, add the exponents:
$$
a^m \times a^n = a^{m+n}
$$

So:
$$
6^2 \times 6^4 = 6^{2+4} = 6^6
$$

✔ Answer: $6^6$

---

Problem 2: Simplify $7^4 \times 7^{-4}$

Again, same base → add exponents:
$$
7^4 \times 7^{-4} = 7^{4 + (-4)} = 7^0
$$

Any non-zero number to the power of 0 is 1:
$$
7^0 = 1
$$

✔ Answer: $1$

---

Problem 3: Simplify $8^5 \div 8^3$

Rule: When dividing powers with the same base, subtract exponents:
$$
a^m \div a^n = a^{m-n}
$$

So:
$$
8^5 \div 8^3 = 8^{5-3} = 8^2
$$

✔ Answer: $8^2$

---

Problem 4: Simplify $\frac{6^5}{6^2}$

Same as above — subtract exponents:
$$
\frac{6^5}{6^2} = 6^{5-2} = 6^3
$$

✔ Answer: $6^3$

---

Problem 5: Simplify $\frac{3^7}{3^3}$

$$
\frac{3^7}{3^3} = 3^{7-3} = 3^4
$$

✔ Answer: $3^4$

---

Problem 6: Simplify $\frac{7^5}{8^3}$

Here, the bases are different (7 and 8), so we cannot combine them using exponent rules.

This expression is already simplified.

✔ Answer: $\frac{7^5}{8^3}$
*(Cannot be simplified further without calculating actual values.)*

---

Problem 7: $(3^3)^4$

Rule: Power of a power → multiply exponents:
$$
(a^m)^n = a^{m \times n}
$$

So:
$$
(3^3)^4 = 3^{3 \times 4} = 3^{12}
$$

✔ Answer: $3^{12}$

---

Problem 8: $(8^2)^3$

$$
(8^2)^3 = 8^{2 \times 3} = 8^6
$$

✔ Answer: $8^6$

---

Problem 9: $(8^3)^4$

$$
(8^3)^4 = 8^{3 \times 4} = 8^{12}
$$

✔ Answer: $8^{12}$

---

Problem 10: $(9^5)^3$

$$
(9^5)^3 = 9^{5 \times 3} = 9^{15}
$$

✔ Answer: $9^{15}$

---

✔ Final Answers:

| Problem | Answer |
|--------|----------------|
| 1 | $6^6$ |
| 2 | $1$ |
| 3 | $8^2$ |
| 4 | $6^3$ |
| 5 | $3^4$ |
| 6 | $\frac{7^5}{8^3}$ |
| 7 | $3^{12}$ |
| 8 | $8^6$ |
| 9 | $8^{12}$ |
| 10 | $9^{15}$ |

---

🔍 Summary of Exponent Rules Used:

1. Product of Powers: $a^m \cdot a^n = a^{m+n}$
2. Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
3. Power of a Power: $(a^m)^n = a^{m \cdot n}$
4. Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

Let me know if you'd like these answers written in expanded form or evaluated numerically!