To solve the equation \( \frac{x^{5}}{2} = 16 x^{\frac{1}{2}} \), we first rewrite the equation for clarity:
\[ x^{\frac{5}{2}} = 16 \cdot x^{\frac{1}{2}} \]
Now, we can divide both sides by \( x^{\frac{1}{2}} \) (assuming \( x \neq 0 \)):
\[ \frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}} = 16 \]
Recall that when dividing exponents with the same base, you subtract the exponents:
\[ x^{\frac{5}{2} - \frac{1}{2}} = 16 \]
This simplifies to:
\[ x^{\frac{4}{2}} = 16 \]
Which further simplifies to:
\[ x^{2} = 16 \]
Next, we take the square root of both sides:
\[ x = \pm 4 \]
So, the solutions to the equation \( \frac{x^{5}}{2} = 16 x^{\frac{1}{2}} \) are:
\[ x = 4 \quad \text{and} \quad x = -4 \]
Since \( x^{\frac{1}{2}} \) (the square root of \( x \)) is defined for non-negative \( x \), the valid solution in the context of real numbers is:
\[ x = 4 \]
Thus, the only relevant solution is \( \boxed{4} \).