The problem requires simplifying the given expression, ensuring all exponents are positive, and evaluating numerical expressions: $\frac{-18x^2y^{-1/2}}{3x^{-1/4}y}$

1. Problem Description

The problem requires simplifying the given expression, ensuring all exponents are positive, and evaluating numerical expressions:

$\frac{-18x^2y^{-1/2}}{3x^{-1/4}y}$

2. Solution Steps

First, simplify the numerical coefficients:

$\frac{-18}{3} = -6$

Next, simplify the expression involving $x$ by using the rule $\frac{x^a}{x^b} = x^{a-b}$:

$\frac{x^2}{x^{-1/4}} = x^{2 - (-1/4)} = x^{2 + 1/4} = x^{\frac{8}{4} + \frac{1}{4}} = x^{\frac{9}{4}}$

Now, simplify the expression involving $y$ by using the rule $\frac{y^a}{y^b} = y^{a-b}$:

$\frac{y^{-1/2}}{y} = \frac{y^{-1/2}}{y^1} = y^{-1/2 - 1} = y^{-1/2 - 2/2} = y^{-3/2}$

Since we need positive exponents, we can rewrite $y^{-3/2}$ as $\frac{1}{y^{3/2}}$.

Combining all the simplified parts gives us:

$-6 x^{\frac{9}{4}} y^{-\frac{3}{2}} = \frac{-6x^{9/4}}{y^{3/2}}$

3. Final Answer

$\frac{-6x^{9/4}}{y^{3/2}}$