Answer
The simplified form of the complex rational expression $\frac{\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}}{\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}}$ is $\frac{3{{x}^{3}}+2{{y}^{3}}}{y\left( x+12 \right)}$.
Work Step by Step
$\frac{\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}}{\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}}$
The numerator and denominator of the complex rational expression are not single, so perform the operation and get a single expression.
The numerator LCD of $\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}$ is $6{{x}^{2}}{{y}^{2}}$, and the denominator LCD of $\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}$ is $6{{x}^{2}}y$.
Solve the expression by making all fractions equivalent as their LCD:
$\begin{align}
& \frac{\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}}{\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}}=\frac{\frac{x}{2{{y}^{2}}}\cdot \frac{6{{x}^{2}}{{y}^{2}}}{6{{x}^{2}}{{y}^{2}}}+\frac{y}{3{{x}^{2}}}\cdot \frac{6{{x}^{2}}{{y}^{2}}}{6{{x}^{2}}{{y}^{2}}}}{\frac{1}{6xy}\cdot \frac{6{{x}^{2}}y}{6{{x}^{2}}y}+\frac{2}{{{x}^{2}}y}\cdot \frac{6{{x}^{2}}y}{6{{x}^{2}}y}} \\
& =\frac{\frac{3{{x}^{3}}}{6{{x}^{2}}{{y}^{2}}}+\frac{2{{y}^{3}}}{6{{x}^{2}}{{y}^{2}}}}{\frac{x}{6{{x}^{2}}y}+\frac{12}{6{{x}^{2}}y}} \\
& =\frac{\frac{3{{x}^{3}}+2{{y}^{3}}}{6{{x}^{2}}{{y}^{2}}}}{\frac{x+12}{6{{x}^{2}}y}}
\end{align}$
Divide the numerator by the denominator,
$\frac{\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}}{\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}}=\frac{3{{x}^{3}}+2{{y}^{3}}}{6{{x}^{2}}{{y}^{2}}}\div \frac{x+12}{6{{x}^{2}}y}$
$\begin{align}
& \frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C} \\
& =\frac{AD}{BC}
\end{align}$
where $\frac{A}{B},\left( B\ne 0 \right)$, $\frac{C}{D},\left( D\ne 0 \right)$ are rational expressions with $\frac{C}{D}\ne 0$
The reciprocal of $\frac{x+12}{6{{x}^{2}}y}$ is $\frac{6{{x}^{2}}y}{x+12}$
So, multiply the reciprocal of the divisor,
$\frac{\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}}{\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}}=\frac{3{{x}^{3}}+2{{y}^{3}}}{6{{x}^{2}}{{y}^{2}}}\cdot \frac{6{{x}^{2}}y}{x+12}$
Simplify the terms,
$\begin{align}
& \frac{\frac{x}{2{{y}^{2}}}+\frac{y}{3{{x}^{2}}}}{\frac{1}{6xy}+\frac{2}{{{x}^{2}}y}}=\frac{\left( 3{{x}^{3}}+2{{y}^{3}} \right)6{{x}^{2}}y}{6{{x}^{2}}{{y}^{2}}\left( x+12 \right)} \\
& =\frac{3{{x}^{3}}+2{{y}^{3}}}{y\left( x+12 \right)}
\end{align}$
