To simplify the expression \((8^{1/2})^3\), we will follow the rules of exponents.
First, we can use the property that \((a^m)^n = a^{m \cdot n}\):
\[ (8^{1/2})^3 = 8^{(1/2) \cdot 3} = 8^{3/2} \]
Next, we can express \(8\) as a power of \(2\):
\[ 8 = 2^3 \]
So we can substitute this into our expression:
\[ 8^{3/2} = (2^3)^{3/2} \]
Again applying the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[ (2^3)^{3/2} = 2^{3 \cdot (3/2)} = 2^{9/2} \]
Now, we can rewrite \(2^{9/2}\) in radical form:
\[ 2^{9/2} = 2^{4 + 1/2} = 2^4 \cdot 2^{1/2} = 16 \cdot \sqrt{2} \]
Thus, the expression \((8^{1/2})^3\) in simplest radical form is:
\[ \boxed{16\sqrt{2}} \]