Answer
The simplest form of the expression $\frac{3{{x}^{4}}{{y}^{6}}{{z}^{-2}}}{-9{{x}^{4}}{{y}^{2}}{{z}^{3}}}$ is $-\frac{{{y}^{4}}}{3{{z}^{5}}}$.
Work Step by Step
Consider the expression.
$\frac{3{{x}^{4}}{{y}^{6}}{{z}^{-2}}}{-9{{x}^{4}}{{y}^{2}}{{z}^{3}}}$
Apply the quotient rule as follows.
$\begin{align}
& \frac{3{{x}^{4}}{{y}^{6}}{{z}^{-2}}}{-9{{x}^{4}}{{y}^{2}}{{z}^{3}}}=\frac{3}{-9}\cdot \frac{{{x}^{4}}}{{{x}^{4}}}\cdot \frac{{{y}^{6}}}{{{y}^{2}}}\cdot \frac{{{z}^{-2}}}{{{z}^{-3}}} \\
& =\frac{3}{-9}{{x}^{4-4}}{{y}^{6-2}}{{z}^{-2-3}} \\
& =-\frac{1}{3}{{x}^{0}}{{y}^{4}}{{z}^{-5}}
\end{align}$
Apply the negative exponent rule,
$\begin{align}
& \frac{3{{x}^{4}}{{y}^{6}}{{z}^{-2}}}{-9{{x}^{4}}{{y}^{2}}{{z}^{3}}}=-\frac{1}{3}{{x}^{0}}{{y}^{4}}{{z}^{-5}} \\
& =-\frac{{{y}^{4}}}{3{{z}^{5}}}
\end{align}$
