## Precalculus (6th Edition) Blitzer

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, $\frac{\frac{1}{x}+\frac{1}{{{x}^{2}}}+\frac{1}{{{x}^{3}}}}{\frac{1}{{{x}^{4}}}+\frac{1}{{{x}^{5}}}+\frac{1}{{{x}^{6}}}}=$ ${{x}^{3}}$.