# 8.4 Building rational expressions and the lcd  (Page 4/4)

 Page 4 / 4

$\begin{array}{lll}\frac{4b}{b-1},\frac{-2b}{b+3}.\hfill & \hfill & \text{By\hspace{0.17em}inspection,\hspace{0.17em}the\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}}\left(b-1\right)\left(b+3\right).\hfill \\ \hfill & \hfill & \text{Rewrite\hspace{0.17em}each\hspace{0.17em}fraction\hspace{0.17em}with\hspace{0.17em}new\hspace{0.17em}denominator\hspace{0.17em}}\left(b-1\right)\left(b+3\right).\hfill \\ \frac{}{\left(b-1\right)\left(b+3\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(b-1\right)\left(b+3\right)}\hfill & \hfill & \begin{array}{l}\text{The\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}first\hspace{0.17em}rational\hspace{0.17em}expression\hspace{0.17em}has\hspace{0.17em}been\hspace{0.17em}multiplied\hspace{0.17em}}\\ \text{by\hspace{0.17em}}b\text{\hspace{0.17em}}+3,\text{\hspace{0.17em}}\text{so\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}4b\text{\hspace{0.17em}must\hspace{0.17em}be\hspace{0.17em}multiplied\hspace{0.17em}by\hspace{0.17em}}b\text{\hspace{0.17em}}+3.\text{\hspace{0.17em}}\end{array}\hfill \\ \hfill & \hfill & 4b\left(b+3\right)=4{b}^{2}+12b\hfill \\ \frac{4{b}^{2}+12b}{\left(b-1\right)\left(b+3\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(b-1\right)\left(b+3\right)}\hfill & \hfill & \begin{array}{l}\text{The\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}second\hspace{0.17em}rational\hspace{0.17em}expression\hspace{0.17em}has\hspace{0.17em}been\hspace{0.17em}multiplied\hspace{0.17em}}\\ \text{by\hspace{0.17em}}b-1\text{,\hspace{0.17em}so\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}-2b\text{\hspace{0.17em}must\hspace{0.17em}be\hspace{0.17em}multiplied\hspace{0.17em}by\hspace{0.17em}}b-1.\end{array}\hfill \\ \hfill & \hfill & -2b\left(b-1\right)=-2{b}^{2}+2b\hfill \\ \frac{4{b}^{2}+12b}{\left(b-1\right)\left(b+3\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{b}^{2}+2b}{\left(b-1\right)\left(b+3\right)}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lll}\frac{6x}{{x}^{2}-8x+15},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{x}^{2}}{{x}^{2}-7x+12}.\hfill & \hfill & \text{We\hspace{0.17em}first\hspace{0.17em}find\hspace{0.17em}the\hspace{0.17em}LCD}.\text{\hspace{0.17em}Factor}.\text{\hspace{0.17em}}\hfill \\ \frac{6x}{\left(x-3\right)\left(x-5\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{x}^{2}}{\left(x-3\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{The\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}}\left(x-3\right)\left(x-5\right)\left(x-4\right).\text{\hspace{0.17em}}\text{Rewrite\hspace{0.17em}each\hspace{0.17em}of\hspace{0.17em}these\hspace{0.17em}}\\ \text{fractions\hspace{0.17em}with\hspace{0.17em}new\hspace{0.17em}denominator\hspace{0.17em}}\left(x-3\right)\left(x-5\right)\left(x-4\right).\end{array}\hfill \\ \frac{}{\left(x-3\right)\left(x-5\right)\left(x-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(x-3\right)\left(x-5\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{By\hspace{0.17em}comparing\hspace{0.17em}the\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}first\hspace{0.17em}fraction\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}LCD\hspace{0.17em}}\\ \text{we\hspace{0.17em}see\hspace{0.17em}that\hspace{0.17em}we\hspace{0.17em}must\hspace{0.17em}multiply\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}6x\text{\hspace{0.17em}by\hspace{0.17em}}x-4.\end{array}\hfill \\ \hfill & \hfill & 6x\left(x-4\right)=6{x}^{2}-24x\hfill \\ \frac{6{x}^{2}-24x}{\left(x-3\right)\left(x-5\right)\left(x-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{}{\left(x-3\right)\left(x-5\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{By\hspace{0.17em}comparing\hspace{0.17em}the\hspace{0.17em}denominator\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}second\hspace{0.17em}fraction\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}LCD,\hspace{0.17em}}\\ \text{we\hspace{0.17em}see\hspace{0.17em}that\hspace{0.17em}we\hspace{0.17em}must\hspace{0.17em}multiply\hspace{0.17em}the\hspace{0.17em}numerator\hspace{0.17em}}-2{x}^{2}\text{\hspace{0.17em}by\hspace{0.17em}}x-5.\end{array}\hfill \\ \hfill & \hfill & -2{x}^{2}\left(x-5\right)=-2{x}^{3}+10{x}^{2}\hfill \\ \hfill & \hfill & \hfill \\ \frac{6{x}^{2}-24x}{\left(x-3\right)\left(x-5\right)\left(x-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{-2{x}^{3}+10{x}^{2}}{\left(x-3\right)\left(x-5\right)\left(x-4\right)}\hfill & \hfill & \hfill \end{array}$

These examples have been done step-by-step and include explanations. This makes the process seem fairly long. In practice, however, the process is much quicker.

$\begin{array}{lll}\frac{6ab}{{a}^{2}-5a+4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{a+b}{{a}^{2}-8a+16}\hfill & \hfill & \hfill \\ \frac{6ab}{\left(a-1\right)\left(a-4\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{a+b}{{\left(a-4\right)}^{2}}\hfill & \hfill & \text{LCD}\text{\hspace{0.17em}}=\left(a-1\right){\left(a-4\right)}^{2}.\hfill \\ \frac{6ab\left(a-4\right)}{\left(a-1\right){\left(a-4\right)}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\left(a+b\right)\left(a-1\right)}{\left(a-1\right){\left(a-4\right)}^{2}}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{x+1}{{x}^{3}+3{x}^{2}},\frac{2x}{{x}^{3}-4x},\frac{x-4}{{x}^{2}-4x+4}\hfill & \hfill & \hfill \\ \frac{x+1}{{x}^{2}\left(x+3\right)},\frac{2x}{x\left(x+2\right)\left(x-2\right)},\frac{x-4}{{\left(x-2\right)}^{2}}\hfill & \hfill & \text{LCD}={x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}.\hfill \end{array}\\ \frac{\left(x+1\right)\left(x+2\right){\left(x-2\right)}^{2}}{{x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}},\frac{2{x}^{2}\left(x+3\right)\left(x-2\right)}{{x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}},\frac{{x}^{2}\left(x+3\right)\left(x+2\right)\left(x-4\right)}{{x}^{2}\left(x+3\right)\left(x+2\right){\left(x-2\right)}^{2}}\end{array}$

## Practice set c

Change the given rational expressions into rational expressions with the same denominators.

$\frac{4}{{x}^{3}},\frac{7}{{x}^{5}}$

$\frac{4{x}^{2}}{{x}^{5}},\frac{7}{{x}^{5}}$

$\frac{2x}{x+6},\frac{x}{x-1}$

$\frac{2x\left(x-1\right)}{\left(x+6\right)\left(x-1\right)},\frac{x\left(x+6\right)}{\left(x+6\right)\left(x-1\right)}$

$\frac{-3}{{b}^{2}-b},\frac{4b}{{b}^{2}-1}$

$\frac{-3\left(b+1\right)}{b\left(b-1\right)\left(b+1\right)},\frac{4{b}^{2}}{b\left(b-1\right)\left(b+1\right)}$

$\frac{8}{{x}^{2}-x-6},\frac{-1}{{x}^{2}+x-2}$

$\frac{8\left(x-1\right)}{\left(x-3\right)\left(x+2\right)\left(x-1\right)},\frac{-1\left(x-3\right)}{\left(x-3\right)\left(x+2\right)\left(x-1\right)}$

$\frac{10x}{{x}^{2}+8x+16},\frac{5x}{{x}^{2}-16}$

$\frac{10x\left(x-4\right)}{{\left(x+4\right)}^{2}\left(x-4\right)},\frac{5x\left(x+4\right)}{{\left(x+4\right)}^{2}\left(x-4\right)}$

$\frac{-2a{b}^{2}}{{a}^{3}-6{a}^{2}},\frac{6b}{{a}^{4}-2{a}^{3}},\frac{-2a}{{a}^{2}-4a+4}$

$\frac{-2{a}^{2}{b}^{2}{\left(a-2\right)}^{2}}{{a}^{3}\left(a-6\right){\left(a-2\right)}^{2}},\frac{6b\left(a-6\right)\left(a-2\right)}{{a}^{3}\left(a-6\right){\left(a-2\right)}^{2}},\frac{-2{a}^{4}\left(a-6\right)}{{a}^{3}\left(a-6\right){\left(a-2\right)}^{2}}$

## Exercises

For the following problems, replace $N$ with the proper quantity.

$\frac{3}{x}=\frac{N}{{x}^{3}}$

$3{x}^{2}$

$\frac{4}{a}=\frac{N}{{a}^{2}}$

$\frac{-2}{x}=\frac{N}{xy}$

$-2y$

$\frac{-7}{m}=\frac{N}{ms}$

$\frac{6a}{5}=\frac{N}{10b}$

$12ab$

$\frac{a}{3z}=\frac{N}{12z}$

$\frac{{x}^{2}}{4{y}^{2}}=\frac{N}{20{y}^{4}}$

$5{x}^{2}{y}^{2}$

$\frac{{b}^{3}}{6a}=\frac{N}{18{a}^{5}}$

$\frac{-4a}{5{x}^{2}y}=\frac{N}{15{x}^{3}{y}^{3}}$

$-12ax{y}^{2}$

$\frac{-10z}{7{a}^{3}b}=\frac{N}{21{a}^{4}{b}^{5}}$

$\frac{8{x}^{2}y}{5{a}^{3}}=\frac{N}{25{a}^{3}{x}^{2}}$

$40{x}^{4}y$

$\frac{2}{{a}^{2}}=\frac{N}{{a}^{2}\left(a-1\right)}$

$\frac{5}{{x}^{3}}=\frac{N}{{x}^{3}\left(x-2\right)}$

$5\left(x-2\right)$

$\frac{2a}{{b}^{2}}=\frac{N}{{b}^{3}-b}$

$\frac{4x}{a}=\frac{N}{{a}^{4}-4{a}^{2}}$

$4ax\left(a+2\right)\left(a-2\right)$

$\frac{6{b}^{3}}{5a}=\frac{N}{10{a}^{2}-30a}$

$\frac{4x}{3b}=\frac{N}{3{b}^{5}-15b}$

$4x\left({b}^{4}-5\right)$

$\frac{2m}{m-1}=\frac{N}{\left(m-1\right)\left(m+2\right)}$

$\frac{3s}{s+12}=\frac{N}{\left(s+12\right)\left(s-7\right)}$

$3s\left(s-7\right)$

$\frac{a+1}{a-3}=\frac{N}{\left(a-3\right)\left(a-4\right)}$

$\frac{a+2}{a-2}=\frac{N}{\left(a-2\right)\left(a-4\right)}$

$\left(a+2\right)\left(a-4\right)$

$\frac{b+7}{b-6}=\frac{N}{\left(b-6\right)\left(b+6\right)}$

$\frac{5m}{2m+1}=\frac{N}{\left(2m+1\right)\left(m-2\right)}$

$5m\left(m-2\right)$

$\frac{4}{a+6}=\frac{N}{{a}^{2}+5a-6}$

$\frac{9}{b-2}=\frac{N}{{b}^{2}-6b+8}$

$9\left(b-4\right)$

$\frac{3b}{b-3}=\frac{N}{{b}^{2}-11b+24}$

$\frac{-2x}{x-7}=\frac{N}{{x}^{2}-4x-21}$

$-2x\left(x+3\right)$

$\frac{-6m}{m+6}=\frac{N}{{m}^{2}+10m+24}$

$\frac{4y}{y+1}=\frac{N}{{y}^{2}+9y+8}$

$4y\left(y+8\right)$

$\frac{x+2}{x-2}=\frac{N}{{x}^{2}-4}$

$\frac{y-3}{y+3}=\frac{N}{{y}^{2}-9}$

${\left(y-3\right)}^{2}$

$\frac{a+5}{a-5}=\frac{N}{{a}^{2}-25}$

$\frac{z-4}{z+4}=\frac{N}{{z}^{2}-16}$

${\left(z-4\right)}^{2}$

$\frac{4}{2a+1}=\frac{N}{2{a}^{2}-5a-3}$

$\frac{1}{3b-1}=\frac{N}{3{b}^{2}+11b-4}$

$b+4$

$\frac{a+2}{2a-1}=\frac{N}{2{a}^{2}+9a-5}$

$\frac{-3}{4x+3}=\frac{N}{4{x}^{2}-13x-12}$

$-3\left(x-4\right)$

$\frac{b+2}{3b-1}=\frac{N}{6{b}^{2}+7b-3}$

$\frac{x-1}{4x-5}=\frac{N}{12{x}^{2}-11x-5}$

$\left(x-1\right)\left(3x+1\right)$

$\frac{3}{x+2}=\frac{3x-21}{N}$

$\frac{4}{y+6}=\frac{4y+8}{N}$

$\left(y+6\right)\left(y+2\right)$

$\frac{-6}{a-1}=\frac{-6a-18}{N}$

$\frac{-8a}{a+3}=\frac{-8{a}^{2}-40a}{N}$

$\left(a+3\right)\left(a+5\right)$

$\frac{y+1}{y-8}=\frac{{y}^{2}-2y-3}{N}$

$\frac{x-4}{x+9}=\frac{{x}^{2}+x-20}{N}$

$\left(x+9\right)\left(x+5\right)$

$\frac{3x}{2-x}=\frac{N}{x-2}$

$\frac{7a}{5-a}=\frac{N}{a-5}$

$-7a$

$\frac{-m+1}{3-m}=\frac{N}{m-3}$

$\frac{k+6}{10-k}=\frac{N}{k-10}$

$-k-6$

For the following problems, convert the given rational expressions to rational expressions having the same denominators.

$\frac{2}{a},\frac{3}{{a}^{4}}$

$\frac{5}{{b}^{2}},\frac{4}{{b}^{3}}$

$\frac{5b}{{b}^{3}},\frac{4}{{b}^{3}}$

$\frac{8}{z},\frac{3}{4{z}^{3}}$

$\frac{9}{{x}^{2}},\frac{1}{4x}$

$\frac{36}{4{x}^{2}},\frac{x}{4{x}^{2}}$

$\frac{2}{a+3},\frac{4}{a+1}$

$\frac{2}{x+5},\frac{4}{x-5}$

$\frac{2\left(x-5\right)}{\left(x+5\right)\left(x-5\right)},\frac{4\left(x+5\right)}{\left(x+5\right)\left(x-5\right)}$

$\frac{1}{x-7},\frac{4}{x-1}$

$\frac{10}{y+2},\frac{1}{y+8}$

$\frac{10\left(y+8\right)}{\left(y+2\right)\left(y+8\right)},\frac{y+2}{\left(y+2\right)\left(y+8\right)}$

$\frac{4}{{a}^{2}},\frac{a}{a+4}$

$\frac{-3}{{b}^{2}},\frac{{b}^{2}}{b+5}$

$\frac{-3\left(b+5\right)}{{b}^{2}\left(b+5\right)},\frac{{b}^{4}}{{b}^{2}\left(b+5\right)}$

$\frac{-6}{b-1},\frac{5b}{4b}$

$\frac{10a}{a-6},\frac{2}{{a}^{2}-6a}$

$\frac{10{a}^{2}}{a\left(a-6\right)},\frac{2}{a\left(a-6\right)}$

$\frac{4}{{x}^{2}+2x},\frac{1}{{x}^{2}-4}$

$\frac{x+1}{{x}^{2}-x-6},\frac{x+4}{{x}^{2}+x-2}$

$\frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)\left(x-3\right)},\frac{\left(x+4\right)\left(x-3\right)}{\left(x-1\right)\left(x+2\right)\left(x-3\right)}$

$\frac{x-5}{{x}^{2}-9x+20},\frac{4}{{x}^{2}-3x-10}$

$\frac{-4}{{b}^{2}+5b-6},\frac{b+6}{{b}^{2}-1}$

$\frac{-4\left(b+1\right)}{\left(b+1\right)\left(b-1\right)\left(b+6\right)},\frac{{\left(b+6\right)}^{2}}{\left(b+1\right)\left(b-1\right)\left(b+6\right)}$

$\frac{b+2}{{b}^{2}+6b+8},\frac{b-1}{{b}^{2}+8b+12}$

$\frac{x+7}{{x}^{2}-2x-3},\frac{x+3}{{x}^{2}-6x-7}$

$\frac{\left(x+7\right)\left(x-7\right)}{\left(x+1\right)\left(x-3\right)\left(x-7\right)},\frac{\left(x+3\right)\left(x-3\right)}{\left(x+1\right)\left(x-3\right)\left(x-7\right)}$

$\frac{2}{{a}^{2}+a},\frac{a+3}{{a}^{2}-1}$

$\frac{x-2}{{x}^{2}+7x+6},\frac{2x}{{x}^{2}+4x-12}$

$\frac{{\left(x-2\right)}^{2}}{\left(x+1\right)\left(x-2\right)\left(x+6\right)},\frac{2x\left(x+1\right)}{\left(x+1\right)\left(x-2\right)\left(x+6\right)}$

$\frac{x-2}{2{x}^{2}+5x-3},\frac{x.-1}{5{x}^{2}+16x+3}$

$\frac{2}{x-5},\frac{-3}{5-x}$

$\frac{2}{x-5},\frac{3}{x-5}$

$\frac{4}{a-6},\frac{-5}{6-a}$

$\frac{6}{2-x},\frac{5}{x-2}$

$\frac{-6}{x-2},\frac{5}{x-2}$

$\frac{k}{5-k},\frac{3k}{k-5}$

$\frac{2m}{m-8},\frac{7}{8-m}$

$\frac{2m}{m-8},\frac{-7}{m-8}$

## Excercises for review

( [link] ) Factor ${m}^{2}{x}^{3}+m{x}^{2}+mx.$

( [link] ) Factor ${y}^{2}-10y+21.$

$\left(y-7\right)\left(y-3\right)$

( [link] ) Write the equation of the line that passes through the points $\left(1,\text{\hspace{0.17em}}1\right)$ and $\left(4,\text{\hspace{0.17em}}-2\right)$ . Express the equation in slope-intercept form.

( [link] ) Reduce $\frac{{y}^{2}-y-6}{y-3}.$

$y+2$

( [link] ) Find the quotient: $\frac{{x}^{2}-6x+9}{{x}^{2}-x-6}÷\frac{{x}^{2}+2x-15}{{x}^{2}+2x}.$

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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right! what he said ⤴⤴⤴
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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