https://file.aiquickdraw.com/mathbot/file/1727112797463cbgg5r1b.jpg

The expression from the image is:

\[
\frac{(10^{-5})^6}{(10^4)^{-8}}
\]

Let's simplify this step by step.

### Step 1: Simplify the powers
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify both the numerator and the denominator.

\[
(10^{-5})^6 = 10^{-30}
\]
\[
(10^4)^{-8} = 10^{4 \cdot (-8)} = 10^{-32}
\]

Now, the expression becomes:

\[
\frac{10^{-30}}{10^{-32}}
\]

### Step 2: Apply division rule of exponents
The rule for dividing exponents with the same base is: 
\[
\frac{a^m}{a^n} = a^{m-n}
\]

So:

\[
\frac{10^{-30}}{10^{-32}} = 10^{-30 - (-32)} = 10^{2}
\]

### Final Answer:
The simplified form of the expression is:

\[
10^2 = 100
\]

Would you like any further details or explanations?

Here are 5 related questions to further your understanding:
1. How would you simplify \(\frac{(10^3)^4}{(10^{-2})^3}\)?
2. Can you explain the difference between multiplying and dividing powers of ten?
3. What happens when you raise a negative exponent to a positive power?
4. How would the answer change if the base of the powers was not 10 but another number, like 5?
5. How do the rules for exponents change when the base is a fraction?

**Tip:** When dealing with exponents, always apply the power rule first before any division or multiplication. This simplifies the expression significantly.

Learn how to simplify the exponential expression (10^{-5})^6 / (10^4)^{-8} step by step. The solution uses key exponent laws, including power of a power and division of exponents. Ideal for students in Grades 8-10.

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