Simplify the expression $[ \frac{1}{(-6x)^{1}y^{5}z^{\frac{1}{6}}} ]^2$ and express the answer with positive exponents.

1. Problem Description

Simplify the expression $[ \frac{1}{(-6x)^{1}y^{5}z^{\frac{1}{6}}} ]^2$ and express the answer with positive exponents.

2. Solution Steps

First, distribute the exponent of 2 to each term inside the parentheses.

$[\frac{1}{(-6x)^{1}y^{5}z^{\frac{1}{6}}}]^2 = \frac{1^2}{[(-6x)^{1}y^{5}z^{\frac{1}{6}}]^2}$

$= \frac{1}{(-6x)^{1*2} y^{5*2} z^{\frac{1}{6}*2}} = \frac{1}{(-6)^{2}x^{2}y^{10}z^{\frac{1}{3}}}$

Since $(-6)^2 = (-6) * (-6) = 36$, the expression becomes

$\frac{1}{36x^{2}y^{10}z^{\frac{1}{3}}}$

3. Final Answer

$\frac{1}{36x^2y^{10}z^{\frac{1}{3}}}$