8.6: Rational Expressions- Answers to the Homework Exercises
- Page ID
- 83243
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reduce Rational Expressions
- \(3\)
- \(\dfrac{1}{2}\)
- undefined; \(a=-1\) is an excluded value
- \(-10\)
- \(0\)
- \(-8,4\)
- \(0,-10\)
- \(0,-\dfrac{1}{2}\)
- \(\dfrac{3}{5a}\)
- \(\dfrac{3m-4}{10}\)
- \(\dfrac{1}{x+7}\)
- \(\dfrac{n+6}{n-5}\)
- \(\dfrac{2x-7}{5x-7}\)
- \(\dfrac{2n-1}{9}\)
- \(\dfrac{9x}{2}\)
- \(\dfrac{1}{9}\)
- \(\dfrac{7r+8}{8r}\)
- \(\dfrac{3(x-3)}{5x+4}\)
- \(\dfrac{9}{p+2}\)
- \(\dfrac{2(m+2)}{5m-3}\)
- \(\dfrac{7n-4}{4}\)
- \(\dfrac{7a+9}{2(3a-2)}\)
- \(\dfrac{5b-8}{5b+2}\)
- \(\dfrac{7x-6}{(3x+4)(x+1)}\)
Multiply and Divide Rational Expressions
- \(4x^2\)
- \(\dfrac{3x^2}{2}\)
- \(\dfrac{r-6}{r+10}\)
- \(\dfrac{x-10}{7}\)
- \(5\)
- \(\dfrac{4(m-5)}{5m^2}\)
- \(\dfrac{b+2}{8b}\)
- \(\dfrac{1}{a+7}\)
- \(9m^2(m+10)\)
- \(\dfrac{5b}{b+5}\)
- \(\dfrac{14}{3}\)
- \(\dfrac{5p}{2}\)
- \(x+4\)
- \(\dfrac{1}{v-10}\)
- \(\dfrac{p-10}{p-4}\)
- \(7\)
- \(\dfrac{v-9}{5}\)
- \(\dfrac{7}{8(k+3)}\)
- \(\dfrac{10}{9(n+6)}\)
- \(n+3\)
- \(\dfrac{3}{2}\)
- \(\dfrac{1}{a+2b}\)
- \(\dfrac{x-3}{5x+2}\)
- \(\dfrac{x+1}{3x+2}\)
Obtain the Lowest Common Denominator
- \(18\)
- \(6a^2c^3\)
- \(x^2-x-12\)
- \(20xy\)
- \(x^2-2x-3\)
- \(12a^4b^5\)
- \((x+2)(x-4)\)
- \((x+1)(x+2)(x+3)\)
- \(4(x-2)\)
- \((x-3)^2(x+3)\)
- \(\dfrac{6a^4}{10a^3b^2},\:\dfrac{2b}{10a^3b^2}\)
- \(\dfrac{x^2-4x}{(x-4)^2(x+4)},\:\dfrac{3x^2+12x}{(x-4)^2(x+4)}\)
- \(\dfrac{4x}{(x-3)(x+2)},\:\dfrac{x^2+4x+4}{(x-3)(x+2)}\)
- \(\dfrac{5}{x(x-6)},\:\dfrac{2x-12}{x(x-6)},\:\dfrac{-3x}{x(x-6)}\)
- \(\dfrac{3x^2+4x+1}{(x-4)(x+3)(x+1)},\:\dfrac{2x^2-8x}{(x-4)(x+3)(x+1)}\)
Add and Subtract Rational Expressions
- \(\dfrac{6}{a+3}\)
- \(\dfrac{x+6}{x-5}\)
- \(\dfrac{15t+16}{18t^3}\)
- \(\dfrac{-7x-13}{4x}\)
- \(\dfrac{-z^2+5z}{z^2-1}\)
- \(\dfrac{4t-5}{4(t-3)}\)
- \(\dfrac{t^2+2ty-y^2}{y^2-t^2}\)
- \(\dfrac{x-8}{(x+8)(x+6)}\)
- \(\dfrac{2x+4}{x^2+4x+3}\)
- \(\dfrac{a-2}{a^2-9}\)
- \(\dfrac{5(x-1)}{(x+1)(x+3)}\)
- \(x-4\)
- \(\dfrac{3x+4}{x^2}\)
- \(\dfrac{5x+9}{24}\)
- \(\dfrac{-c^2+cd-d^2}{c^2d^2}\)
- \(\dfrac{11x+15}{4x(x+5)}\)
- \(\dfrac{2x+10}{(x+3)^2}\)
- \(\dfrac{2x^2-10x+25}{x(x-5)}\)
- \(\dfrac{2x-5}{(x-3)(x-2)}\)
- \(\dfrac{2x+7}{x^2+5x+6}\)
- \(\dfrac{2}{y^2-y}\)
- \(\dfrac{5x+5}{x^2+2x-15}\)
- \(\dfrac{x+1}{x+2}\)
- \(\dfrac{x-4}{x+2}\)
Compound Rational Expressions
- \(\dfrac{x}{x-1}\)
- \(-\dfrac{a-1}{a+1}\)
- \(-\dfrac{1}{2}\)
- \(\dfrac{x}{3}\)
- \(\dfrac{x-5}{x+9}\)
- \(\dfrac{x-2}{x+2}\)
- \(\dfrac{5-a}{a}\)
- \(\dfrac{4}{5}\)
- \(\dfrac{2a^2-3a+3}{-4a^2-2a}\)
- \(\dfrac{x+2}{x-1}\)
- \(\dfrac{1}{x+4}\)
- \(\dfrac{x-3}{x+4}\)
- \(\dfrac{a-3b}{a+3b}\)
- \(\dfrac{-2a-2}{3a-4}\)
- \(-\dfrac{2x}{x^2+1}\)