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The problem provided asks to find the value of \( \frac{1}{4^{a \cdot b}} \), given the following two equations:

1. \( 125^a = 9 \)
2. \( 3^b = 8 \)

### Step 1: Solve for \( a \)
From \( 125^a = 9 \), we know that \( 125 = 5^3 \). Therefore, we can rewrite the equation as:

\[
(5^3)^a = 9
\]

This simplifies to:

\[
5^{3a} = 9
\]

Taking the logarithm (base 5) of both sides:

\[
3a = \log_5 9
\]

Using the change of base formula \( \log_b x = \frac{\log x}{\log b} \), we get:

\[
3a = \frac{\log 9}{\log 5}
\]

Thus, 

\[
a = \frac{\log 9}{3 \log 5}
\]

### Step 2: Solve for \( b \)
From \( 3^b = 8 \), we take the logarithm (base 3) of both sides:

\[
b = \log_3 8
\]

Using the change of base formula:

\[
b = \frac{\log 8}{\log 3}
\]

### Step 3: Calculate \( 4^{a \cdot b} \)
We need to find \( \frac{1}{4^{a \cdot b}} \). Since \( 4 = 2^2 \), this becomes:

\[
4^{a \cdot b} = (2^2)^{a \cdot b} = 2^{2a \cdot b}
\]

So, \( \frac{1}{4^{a \cdot b}} = 2^{-2a \cdot b} \).

Now, let's calculate \( a \cdot b \). Using the expressions for \( a \) and \( b \):

\[
a \cdot b = \left( \frac{\log 9}{3 \log 5} \right) \cdot \left( \frac{\log 8}{\log 3} \right)
\]

Simplifying this will give us the final answer. I'll compute this next.The value of \( \frac{1}{4^{a \cdot b}} \) is approximately \( 0.303 \), which is close to \( \frac{1}{3} \). Therefore, the closest answer is:

\[
\boxed{C) 3}
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How does the change of base formula for logarithms work?
2. Why do we use logarithms to solve exponential equations?
3. How does simplifying powers help in solving complex equations?
4. Can we solve similar problems using natural logarithms?
5. How would the solution change if the base numbers (125 or 3) were different?

**Tip:** Using the properties of logarithms and change of base formulas can simplify otherwise complex-looking problems, making them easier to solve.

Given 125^a = 9 and 3^b = 8, what is the value of 1 / 4^(a * b)?

Learn how to solve the exponential equation involving 125^a = 9 and 3^b = 8 to find the value of 1 / 4^(a * b). This solution covers key algebraic concepts, including logarithms, exponent rules, and the change of base formula. Suitable for students in Grades 10-12.

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