8.8 Complex rational expressions

$\begin{array}{l}\frac{\frac{x^3}{8}}{\frac{x^5}{12}}\\ \\ \text{Steps\hspace{0.17em}1\hspace{0.17em}and\hspace{0.17em}2\hspace{0.17em}are\hspace{0.17em}not\hspace{0.17em}necessary\hspace{0.17em}so\hspace{0.17em}we\hspace{0.17em}proceed\hspace{0.17em}with\hspace{0.17em}step\hspace{0.17em}3}\text{.}\\ \\ \frac{\frac{x^3}{8}}{\frac{x^5}{12}}=\frac{x^3}{8}·\frac{12}{x^5}=\frac{\cancel{x^3}}{\underset{2}{\cancel{8}}}·\frac{\stackrel{3}{\cancel{12}}}{x^{\cancel{5}2}}=\frac{3}{2x^2}\end{array}$

$\begin{array}{l}\frac{1-\frac{1}{x}}{1-\frac{1}{x^2}}\\ \\ \text{Step\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}numerator:\hspace{0.17em}LCD}=x\text{.}\\ 1-\frac{1}{x}=\frac{x}{x}-\frac{1}{x}=\frac{x-1}{x}\\ \\ \text{Step\hspace{0.17em}2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}denominator:\hspace{0.17em}LCD}=x^2\\ \\ 1-\frac{1}{x^2}=\frac{x^2}{x^2}-\frac{1}{x^2}=\frac{x^2-1}{x^2}\\ \\ \text{Step\hspace{0.17em}3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}by\hspace{0.17em}the\hspace{0.17em}denominator}\text{.}\\ \\ \begin{array}{lll}\frac{\frac{x-1}{x}}{\frac{x^2-1}{x^2}}\hfill & =\hfill & \frac{x-1}{x}·\frac{x^2}{x^2-1}\hfill \\ \hfill & =\hfill & \frac{\cancel{x-1}}{\cancel{x}}\frac{x^{\cancel{2}}}{\left(x+1\right)\cancel{\left(x-1\right)}}\hfill \\ \hfill & =\hfill & \frac{x}{x+1}\hfill \end{array}\\ \\ \text{Thus,}\\ \\ \frac{1-\frac{1}{x}}{1-\frac{1}{x^2}}=\frac{x}{x+1}\end{array}$

$\begin{array}{l}\frac{2-\frac{13}{m}-\frac{7}{m^2}}{2+\frac{3}{m}+\frac{1}{m^2}}\\ \\ \text{Step\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}numerator:\hspace{0.17em}LCD}=m^2\text{.}\\ \\ 2-\frac{13}{m}-\frac{7}{m^2}=\frac{2m^2}{m^2}-\frac{13m}{m^2}-\frac{7}{m^2}=\frac{2m^2-13m-7}{m^2}\\ \\ \text{Step\hspace{0.17em}2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Combine\hspace{0.17em}the\hspace{0.17em}terms\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}denominator:\hspace{0.17em}LCD}=m^2\\ \\ 2+\frac{3}{m}+\frac{1}{m^2}=\frac{2m^2}{m^2}+\frac{3m}{m^2}+\frac{1}{m^2}=\frac{2m^2+3m+1}{m^2}\\ \\ \text{Step\hspace{0.17em}3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}by\hspace{0.17em}the\hspace{0.17em}denominator}\text{.}\\ \\ \begin{array}{lll}\frac{\frac{2m^2-13m-7}{m^2}}{\frac{2m^2+3m-1}{m^2}}\hfill & =\hfill & \frac{2m^2-13m-7}{m^2}·\frac{m^2}{2m^2+3m+1}\hfill \\ \hfill & =\hfill & \frac{\cancel{\left(2m+1\right)}\left(m-7\right)}{\cancel{m^2}}·\frac{\cancel{m^2}}{\cancel{\left(2m+1\right)}\left(m+1\right)}\hfill \\ \hfill & =\hfill & \frac{m-7}{m+1}\hfill \end{array}\\ \\ \text{Thus,}\\ \\ \frac{2-\frac{13}{m}-\frac{7}{m^2}}{2+\frac{3}{m}+\frac{1}{m^2}}=\frac{m-7}{m+1}\end{array}$

$$\begin{array}{l}\frac{1-\frac{4}{a^2}}{1+\frac{2}{a}}\\ \\ \text{Step\hspace{0.17em}1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}The\hspace{0.17em}LCD}=a^2\text{.}\\ \text{Step\hspace{0.17em}2:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Multiply\hspace{0.17em}both\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}and\hspace{0.17em}denominator\hspace{0.17em}by\hspace{0.17em}}{a}^2\text{.}\\ \\ \begin{array}{lll}\frac{a^2\left(1-\frac{4}{a^2}\right)}{a^2\left(1+\frac{2}{a}\right)}\hfill & =\hfill & \frac{a^2·1-a^2·\frac{4}{a^2}}{a^2·1+a^2·\frac{2}{a}}\hfill \\ \hfill & =\hfill & \frac{a^2-4}{a^2+2a}\hfill \end{array}\\ \\ \text{Step\hspace{0.17em}3:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Reduce}\text{.}\\ \\ \begin{array}{lll}\frac{a^2-4}{a^2+2a}\hfill & =\hfill & \frac{\cancel{\left(a+2\right)}\left(a-2\right)}{a\cancel{\left(a+2\right)}}\hfill \\ \hfill & =\hfill & \frac{a-2}{a}\hfill \end{array}\\ \\ \text{Thus,}\\ \\ \frac{1-\frac{4}{a^2}}{1+\frac{2}{a}}=\frac{a-2}{a}\end{array}$$

$$\begin{array}{l}\begin{array}{l}\frac{1-\frac{5}{x}-\frac{6}{x^2}}{1+\frac{6}{x}+\frac{5}{x^2}}\hfill \\ \hfill \\ \text{Step}1:\text{The}\text{LCD}\text{is}{x}^2.\hfill \\ \text{Step}2:\text{Multiply}\text{the}\text{numerator}\text{and}\text{denominator}\text{by}{x}^2.\hfill \\ \hfill \\ \begin{array}{lll}\frac{x^2(1-\frac{5}{x}-\frac{6}{x^2})}{x^2(1+\frac{6}{x}+\frac{5}{x^2})}\hfill & =\hfill & \frac{x^2·1-x^{\cancel{2}}·\frac{5}{\cancel{x}}-\cancel{x^2}·\frac{6}{\cancel{x^2}}}{x^2·1+x^{\cancel{2}}·\frac{6}{\cancel{x}}+\cancel{x^2}·\frac{5}{\cancel{x^2}}}\hfill \\ \hfill & =\hfill & \frac{x^2-5x-6}{x^2+6x+5}\hfill \end{array}\hfill \end{array}\\ \text{Step}3:\text{Reduce}\text{.}\\ \begin{array}{lll}\frac{x^2-5x-6}{x^2+6x+5}\hfill & =\hfill & \frac{\left(x-6\right)\left(x+1\right)}{\left(x+5\right)\left(x+1\right)}\hfill \\ \hfill & =\hfill & \frac{x-6}{x+5}\hfill \end{array}\\ \text{Thus,}\\ \frac{1-\frac{5}{x}-\frac{6}{x^2}}{1+\frac{6}{x}+\frac{5}{x^2}}=\frac{x-6}{x+5}\end{array}$$