Simplify the expression $\frac{-36x^2y^{-1/4}}{6x^{-1/5}y}$ and express the answer with positive exponents.

1. Problem Description

Simplify the expression $\frac{-36x^2y^{-1/4}}{6x^{-1/5}y}$ and express the answer with positive exponents.

2. Solution Steps

First, divide the coefficients: $\frac{-36}{6} = -6$.

Next, simplify the $x$ terms using the quotient rule for exponents:

$\frac{x^2}{x^{-1/5}} = x^{2 - (-1/5)} = x^{2 + 1/5} = x^{\frac{10}{5} + \frac{1}{5}} = x^{11/5}$.

Now, simplify the $y$ terms using the quotient rule for exponents:

$\frac{y^{-1/4}}{y} = y^{-1/4 - 1} = y^{-1/4 - 4/4} = y^{-5/4}$.

To express the answer with positive exponents, we rewrite $y^{-5/4}$ as $\frac{1}{y^{5/4}}$.

Combining these results, we have:

$-6 x^{11/5} y^{-5/4} = -6 x^{11/5} \frac{1}{y^{5/4}} = \frac{-6x^{11/5}}{y^{5/4}}$.

3. Final Answer

The simplified expression is $\frac{-6x^{11/5}}{y^{5/4}}$.