Answer
The required solution is:
It cubes x.
Work Step by Step
We have the given rational expression:
$\frac{\frac{1}{x}+\frac{1}{{{x}^{2}}}+\frac{1}{{{x}^{3}}}}{\frac{1}{{{x}^{4}}}+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{6}}}}$
Solve the numerator of the given expression:
$\begin{align}
& \frac{1}{x}+\frac{1}{{{x}^{2}}}+\frac{1}{{{x}^{3}}}=\frac{1}{x}\times \frac{{{x}^{2}}}{{{x}^{2}}}+\frac{1}{{{x}^{2}}}\times \frac{x}{x}+\frac{1}{{{x}^{3}}} \\
& =\frac{{{x}^{2}}}{{{x}^{3}}}+\frac{x}{{{x}^{3}}}+\frac{1}{{{x}^{3}}} \\
& =\frac{{{x}^{2}}+x+1}{{{x}^{3}}}
\end{align}$
Also, solve the denominator of the given expression:
$\begin{align}
& \frac{1}{{{x}^{4}}}+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{6}}}=\frac{1}{{{x}^{4}}}\times \frac{{{x}^{2}}}{{{x}^{2}}}+\frac{1}{{{x}^{5}}}\times \frac{x}{x}+\frac{1}{{{x}^{6}}} \\
& =\frac{{{x}^{2}}}{{{x}^{4}}}+\frac{x}{{{x}^{5}}}+\frac{1}{{{x}^{6}}} \\
& =\frac{{{x}^{2}}+x+1}{{{x}^{6}}}
\end{align}$
So, the given rational expression takes the form:
$\frac{\frac{1}{x}+\frac{1}{{{x}^{2}}}+\frac{1}{{{x}^{3}}}}{\frac{1}{{{x}^{4}}}+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{6}}}}=\frac{\frac{{{x}^{2}}+x+1}{{{x}^{3}}}}{\frac{{{x}^{2}}+x+1}{{{x}^{6}}}}$
And simplify the above complex rational expression.
$\begin{align}
& \frac{\frac{1}{x}+\frac{1}{{{x}^{2}}}+\frac{1}{{{x}^{3}}}}{\frac{1}{{{x}^{4}}}+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{6}}}}=\frac{{{x}^{2}}+x+1}{{{x}^{3}}}\times \frac{{{x}^{6}}}{{{x}^{2}}+x+1} \\
& =\frac{{{x}^{6}}}{{{x}^{3}}} \\
& ={{x}^{6-3}} \\
& ={{x}^{3}}
\end{align}$
Hence, $\frac{\frac{1}{x}+\frac{1}{{{x}^{2}}}+\frac{1}{{{x}^{3}}}}{\frac{1}{{{x}^{4}}}+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{6}}}}=$ ${{x}^{3}}$.
