The problem asks to evaluate the expression $(\frac{1}{2})^{3^2} \cdot (\frac{1}{2})^3$.

1. Problem Description

The problem asks to evaluate the expression $(\frac{1}{2})^{3^2} \cdot (\frac{1}{2})^3$.

2. Solution Steps

We need to simplify the expression $(\frac{1}{2})^{3^2} \cdot (\frac{1}{2})^3$.

First, we evaluate $3^2 = 9$.

Then the expression becomes $(\frac{1}{2})^9 \cdot (\frac{1}{2})^3$.

Using the rule $a^m \cdot a^n = a^{m+n}$, we have $(\frac{1}{2})^9 \cdot (\frac{1}{2})^3 = (\frac{1}{2})^{9+3} = (\frac{1}{2})^{12}$.

Now we evaluate $(\frac{1}{2})^{12} = \frac{1^{12}}{2^{12}} = \frac{1}{2^{12}}$.

Since $2^{10} = 1024$, we have $2^{12} = 2^{10} \cdot 2^2 = 1024 \cdot 4 = 4096$.

Therefore, $(\frac{1}{2})^{12} = \frac{1}{4096}$.

$a^m \cdot a^n = a^{m+n}$

3. Final Answer

The final answer is $\frac{1}{4096}$.