We are given the following quotient to simplify:

$\frac{6.4 \times 10^{-8}}{8 \times 10^{-4}}$

Step 1: Separate the constants and the powers of 10

We can simplify this expression by handling the constants and powers of 10 separately.

$\frac{6.4}{8} \times \frac{10^{-8}}{10^{-4}}$

Step 2: Simplify the constants

Now, simplify the constant fraction:

$\frac{6.4}{8} = 0.8$

Step 3: Simplify the powers of 10

For the powers of 10, use the rule for division of exponents:

$\frac{10^{-8}}{10^{-4}} = 10^{-8 - (-4)} = 10^{-8 + 4} = 10^{-4}$

Step 4: Combine the results

Now, multiply the results from Step 2 and Step 3:

$0.8 \times 10^{-4}$

Step 5: Express in scientific notation

To express $0.8 \times 10^{-4}$ in scientific notation, we rewrite $0.8$ as $8 \times 10^{-1}$. So, the expression becomes:

$8 \times 10^{-1} \times 10^{-4} = 8 \times 10^{-5}$

Thus, the simplified quotient in scientific notation is:

$8 \times 10^{-5}$

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Would you like further details or have any questions on this? Here are some related questions for you to explore:

1. What is the rule for dividing powers of 10?
2. How can we convert a number like 0.08 into scientific notation?
3. What are the steps for simplifying a product of scientific notation expressions?
4. How do you add and subtract numbers in scientific notation?
5. How can you multiply numbers in scientific notation?

Tip: When multiplying or dividing numbers in scientific notation, always handle the constants and the powers of 10 separately.