Let's simplify the left side of the equation step by step.
First, we start with the left-hand side:
\[ 4^3 \cdot (64^{1/2})^3 \]
We can simplify \(64^{1/2}\):
\[ 64^{1/2} = 8 \]
Therefore, we rewrite the equation as:
\[ 4^3 \cdot 8^3 \]
Next, we can express \(8\) in terms of \(4\):
\[ 8 = 2^3 = (4^{1/2})^3 = 4^{3/2} \]
So now we can express \(8^3\) as:
\[ 8^3 = (4^{3/2})^3 = 4^{9/2} \]
Now, we can substitute that back into our equation:
\[ 4^3 \cdot 8^3 = 4^3 \cdot 4^{9/2} \]
Using the laws of exponents (adding the exponents), we have:
\[ 4^{3 + 9/2} \]
To add \(3\) and \(9/2\), we first convert \(3\) to a fraction:
\[ 3 = \frac{6}{2} \]
So we have:
\[ 3 + \frac{9}{2} = \frac{6}{2} + \frac{9}{2} = \frac{15}{2} \]
Thus, the left-hand side simplifies to:
\[ 4^{15/2} \]
Now we look at the right-hand side \(4^6\) and adjust it for comparison:
\[ 4^6 = 4^{12/2} \]
Now we see:
\[ 4^{15/2} \neq 4^{12/2} \]
Therefore, since \(4^{15/2}\) does not equal \(4^{12/2}\), the original equation is false.
Thus, the answer is:
\[ \boxed{2} \]
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