To solve the expression \( \frac{4^{111}}{4^{108}} \), we can use the properties of exponents. 

### Step-by-Step Solution:

1. **Apply the Quotient Rule for Exponents**:
   \[
   \frac{a^m}{a^n} = a^{m-n}
   \]
   In this case, \(a = 4\), \(m = 111\), and \(n = 108\).

2. **Simplify the Expression**:
   \[
   \frac{4^{111}}{4^{108}} = 4^{111 - 108}
   \]

3. **Calculate the Exponent**:
   \[
   111 - 108 = 3
   \]

4. **Simplify Further**:
   \[
   4^{111 - 108} = 4^3
   \]

5. **Compute \(4^3\)**:
   \[
   4^3 = 4 \times 4 \times 4 = 64
   \]

### Final Answer:
\[
\frac{4^{111}}{4^{108}} = 64
\]

Would you like more details on this process, or do you have any other questions?

Here are five related questions that might help expand your understanding:

1. How do you apply the properties of exponents to simplify expressions with different bases?
2. What is the result of \( \frac{5^{9}}{5^{6}} \)?
3. How does the quotient rule for exponents change if the bases are not the same?
4. How would you compute \( \frac{9^{7}}{9^{2}} \)?
5. What is the significance of negative exponents, and how are they interpreted?

**Tip:** Always remember that when dividing powers with the same base, you subtract the exponents!

Learn how to simplify and solve the expression \( \frac{4^{111}}{4^{108}} \) using the properties of exponents. This problem demonstrates the application of the quotient rule for exponents and involves calculating powers with the same base. Suitable for students in Grades 7-9.

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