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Let's analyze the given problem step by step:

The question gives us the following information:
1. \( 125^a = 9 \)
2. \( 3^b = 8 \)

We are asked to find the value of:

\[
4^{\frac{a}{2}} \cdot b
\]

### Step 1: Solving for \( a \)
We know that \( 125 = 5^3 \), so we can rewrite \( 125^a = 9 \) as:

\[
(5^3)^a = 9 \quad \Rightarrow \quad 5^{3a} = 9
\]

Taking the natural logarithm of both sides:

\[
3a \cdot \ln(5) = \ln(9)
\]

Solving for \( a \):

\[
a = \frac{\ln(9)}{3 \ln(5)}
\]

### Step 2: Solving for \( b \)
We know that \( 3^b = 8 \), so by taking the natural logarithm of both sides:

\[
b \cdot \ln(3) = \ln(8)
\]

Solving for \( b \):

\[
b = \frac{\ln(8)}{\ln(3)}
\]

### Step 3: Substituting \( a \) and \( b \) into the expression
We are tasked with finding \( 4^{\frac{a}{2}} \cdot b \). First, we rewrite 4 as \( 2^2 \), so:

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

Thus, the expression becomes:

\[
2^a \cdot b
\]

Now, we need to plug in the values of \( a \) and \( b \) from Steps 1 and 2.

### Step 4: Approximation and final result
We can calculate the approximations for \( a \) and \( b \), and then substitute them into the final expression.

I'll now calculate the values.The value of the expression \( 4^{\frac{a}{2}} \cdot b \) is approximately \( 2.59 \), which is closest to option **D) 2**.

Would you like further details on the steps? Here are 5 related questions you might find interesting:

1. How does logarithmic manipulation help in solving exponential equations?
2. Can you explain the approximation methods used for logarithms?
3. What is the significance of the natural logarithm in calculus and algebra?
4. How do exponent properties simplify complex expressions?
5. Why is it important to approximate logarithmic values for real-world applications?

**Tip:** Always remember that changing the base of logarithms is a powerful tool when solving complex exponential equations.

125^a = 9, 3^b = 8 olduğuna göre, 4^(a/2) * b değeri kaçtır?

Learn how to solve the equation 125^a = 9 and 3^b = 8 to find the value of 4^(a/2) * b. This problem involves logarithmic manipulation and exponent rules, suitable for high school students in Grades 10-12.

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