We are asked to find the expression equivalent to:

$\frac{x^3}{\sqrt{x}}$

We can simplify this expression by rewriting the square root in terms of an exponent. Recall that:

$\sqrt{x} = x^{1/2}$

Thus, the original expression becomes:

$\frac{x^3}{x^{1/2}}$

To simplify this, we subtract the exponent in the denominator from the exponent in the numerator:

$x^{3 - 1/2} = x^{\frac{6}{2} - \frac{1}{2}} = x^{\frac{5}{2}}$

So, the correct equivalent expression is:

$\boxed{x^{\frac{5}{2}}}$

This corresponds to option C.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you handle negative exponents in similar expressions?
2. What is the general rule for dividing powers with the same base?
3. How would the solution change if the denominator was $\sqrt[3]{x}$ instead of $\sqrt{x}$?
4. Can you apply the same method if the variable $x$ is raised to a fractional power in both numerator and denominator?
5. How does this method extend to expressions with multiple variables?

Tip: Always remember that square roots are equivalent to raising a number to the power of $\frac{1}{2}$. This can help simplify expressions involving roots and exponents.