The problem asks to rewrite $\frac{4^{-3}}{4^5}$ using a single positive exponent. Let’s solve this step-by-step.

Step 1: Apply the quotient rule for exponents

The quotient rule states: $\frac{a^m}{a^n} = a^{m-n}.$

For $\frac{4^{-3}}{4^5}$, this becomes: $4^{-3 - 5}.$

Step 2: Simplify the exponent

$-3 - 5 = -8.$

Thus, the expression becomes: $4^{-8}.$

Step 3: Convert to a positive exponent

To express $4^{-8}$ with a positive exponent, recall the rule: $a^{-n} = \frac{1}{a^n}.$

So: $4^{-8} = \frac{1}{4^8}.$

Final Answer:

$\frac{1}{4^8}.$

Would you like additional clarification or details? Here are 5 related questions to explore further:

1. What is the general rule for simplifying expressions with negative exponents?
2. How do we simplify expressions with multiple variables and exponents?
3. Can you explain how to rewrite $x^{-m} y^{-n}$ in terms of positive exponents?
4. What is the expanded value of $4^8$, and how do we compute it?
5. How does the quotient rule apply when the base values differ?

Tip: Always simplify exponents step by step, focusing on applying one rule at a time.