To solve the expression:

$\frac{4.16 \times 10^{-5}}{8.0 \times 10^{-4}}$

Follow these steps:

1. Rewrite the expression:

$\frac{4.16 \times 10^{-5}}{8.0 \times 10^{-4}} = \frac{4.16}{8.0} \times \frac{10^{-5}}{10^{-4}}$

2. Simplify the fraction $\frac{4.16}{8.0}$:

$\frac{4.16}{8.0} = 0.52$

3. Simplify the powers of 10:

$\frac{10^{-5}}{10^{-4}} = 10^{-5 + 4} = 10^{-1}$

4. Combine the results:

$0.52 \times 10^{-1} = 0.052$

Thus, the value of the expression is:

$\boxed{0.052}$

Would you like more details or have any questions? Here are five related questions to explore further:

1. How do you simplify expressions with scientific notation?
2. What is the rule for dividing powers of 10?
3. How do you multiply decimals by powers of 10?
4. Can you show how to convert $4.16 \times 10^{-5}$ into decimal form?
5. How do you compare two numbers written in scientific notation?

Tip: When dividing numbers in scientific notation, always simplify the coefficients and handle the exponents separately!