Answer
The simplified form of the rational expression is $\frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}$.
The combined resistance when ${{R}_{1}}=4\text{ ohms, }{{R}_{2}}=8\text{ ohms }{{R}_{3}}=12\text{ ohms}$ is $\frac{24}{11}\text{ohms}$.
Work Step by Step
Consider the expression, $\frac{1}{\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}}}$
Therefore,
$\begin{align}
& \frac{1}{\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}}}=\frac{1}{\frac{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}{{{R}_{1}}{{R}_{2}}{{R}_{3}}}} \\
& =\frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}
\end{align}$
Thus, the simplified form of the provided expression is $\frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}$
Now, substitute the value ${{R}_{1}}=4\text{ ohms, }{{R}_{2}}=8\text{ ohms }{{R}_{3}}=12\text{ ohms}$ in the expression,
$\frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}$
Therefore,
$\begin{align}
& \frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}=\frac{4\cdot 8\cdot 12}{8\cdot 12+4\cdot 12+4\cdot 8} \\
& =\frac{384}{176} \\
& =\frac{24}{11}
\end{align}$
The simplified form of the rational expression is $\frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{2}}{{R}_{3}}+{{R}_{1}}{{R}_{3}}+{{R}_{1}}{{R}_{2}}}$.
The combined resistance when ${{R}_{1}}=4\text{ ohms, }{{R}_{2}}=8\text{ ohms }{{R}_{3}}=12\text{ ohms}$ is $\frac{24}{11}\text{ohms}$.
