Let's simplify the given expression:

$\frac{4^8 + 2^{20}}{16^5}$

Step 1: Express everything in terms of powers of 2.

We know: $4 = 2^2 \quad \text{and} \quad 16 = 2^4.$

Thus, we can rewrite each term:

• $4^8 = (2^2)^8 = 2^{16}$,
• $16^5 = (2^4)^5 = 2^{20}$.

Now the expression becomes: $\frac{2^{16} + 2^{20}}{2^{20}}.$

Step 2: Simplify the numerator.

Factor out $2^{16}$ from the numerator: $2^{16} + 2^{20} = 2^{16}(1 + 2^4).$ Since $2^4 = 16$, we have: $2^{16}(1 + 16) = 2^{16} \times 17.$

So, the expression becomes: $\frac{2^{16} \times 17}{2^{20}}.$

Step 3: Simplify the fraction.

We can cancel out $2^{16}$ from both the numerator and denominator: $\frac{2^{16} \times 17}{2^{20}} = 17 \times \frac{2^{16}}{2^{20}} = 17 \times 2^{16 - 20} = 17 \times 2^{-4}.$

Step 4: Final expression.

Now we simplify $2^{-4}$ as: $2^{-4} = \frac{1}{2^4} = \frac{1}{16}.$

Thus, the final result is: $17 \times \frac{1}{16} = \frac{17}{16}.$

Final Answer:

$\frac{17}{16}.$

Would you like any further details or clarifications on the steps? Here are some related questions you might find interesting:

1. How do we apply the properties of exponents when simplifying expressions like this one?
2. What is the importance of factoring in simplifying exponents?
3. How does one approach simplifying expressions with different bases in the numerator and denominator?
4. What are some common mistakes when dealing with powers of 2 in algebraic expressions?
5. How would this problem change if the denominator was $2^{10}$ instead of $16^5$?

Tip: Always try to express numbers in terms of powers of a common base (like 2) to make exponentiation problems simpler!