Rational Expressions and Functions

Find the domain of $R\left(x\right)=\frac{4x^2-16x}{8x^2-16x-64}$.

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{3}+\frac{1}{{\mathrm{x}}^2}>\frac{4}{3\mathrm{x}}$

Francesca is $5.75$ feet tall. Late one afternoon, her shadow was $8$ feet long. At the same time, the shadow of a nearby tree was $32$ ft long. Find the height of the tree.

Determine the value for which each rational expression is undefined.

$$\begin{gathered} \left(\text{a}\right)\frac{3y^2}{8x} \\ \left(\text{b}\right)\frac{8n-5}{3n+1} \\ \left(\text{c}\right)\frac{a+10}{a^2+4a+3} \end{gathered}$$

Determine the value for which each rational expression is undefined.

$$\begin{gathered} \left(\text{a}\right)\frac{4p}{5q} \\ \left(\text{b}\right)\frac{y-1}{3y+2} \\ \left(\text{c}\right)\frac{m-5}{m^2+m-6} \end{gathered}$$

Simplify:$\frac{x^2-x-2}{x^2-3x+2}$

Simplify: $\frac{x^2-3x-10}{x^2+x-2}$

Simplify $\frac{2x^2-12xy+18y^2}{3x^2-27y^2}$.

Simplify $\frac{5x^2-30xy+25y^2}{2x^2-50y^2}.$

Simplify $\frac{x^2-4x-5}{25-x^2}$

Simplify$\frac{x^2+x-2}{1-x^2}.$

Simplify $\frac{5x}{x^2+5x+6}·\frac{x^2-4}{10x}$.

Simplify $\frac{9x^2}{x^2+11x+30}·\frac{x^2-36}{3x^2}$.

Simplify $\frac{2x^2+5x-12}{x^2-16}·\frac{x^2-8x+16}{2x^2-13x+15}$.

Simplify $\frac{x^3-8}{3x^2-6x+12}÷\frac{x^2-4}{6}$

Simplify $\frac{2z^2}{z^2-1}÷\frac{z^3-z^2+z}{z^3+1}$

Simplify $\frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}$.

Simplify $\frac{{\displaystyle \frac{y^2-36}{2y^2+11y-6}}}{{\displaystyle \frac{2y^2-2y-60}{8y-4}}}$.

Perform the indicated operations.

$\frac{4m+4}{3m-15}·\frac{m^2-3m-10}{m^2-4m-32}÷\frac{12m-36}{6m-48}.$

Perform the indicated operations.

$\frac{2n^2+10n}{n-1}÷\frac{n^2+10n+24}{n^2+8n-9}·\frac{n+4}{8n^2+12n}$.

Perform the indicated operation $\frac{2n^2+10n}{n-1}÷\frac{n^2+10n+24}{n^2+8n-9}·\frac{n+4}{8n^2+12n}$

Find the domain of $R\left(x\right)=\frac{2x^2-10x}{4x^2-16x-20}$.

Find $R\left(x\right)=f\left(x\right)·g\left(x\right)$ where $f\left(x\right)=\frac{3x-21}{x^2-9x+14}$and $g\left(x\right)=\frac{2x^2-8}{3x+6}$.

Find $R\left(x\right)=f\left(x\right)·g\left(x\right)$ where $f\left(x\right)=\frac{x^2-x}{3x^2+27x-30}$and $g\left(x\right)=\frac{x^2-100}{x^2-10x}.$

Find $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ where $f\left(x\right)=\frac{2x^2}{x^2-8x}$and $g\left(x\right)=\frac{8x^2+24x}{x^2+x-6}.$

Find $R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$where and $f\left(x\right)=\frac{15x^2}{3x^2+33x}$and $g\left(x\right)=\frac{5x-5}{x^2+9x-22}.$

In the following exercises, determine the values for which the rational expression is defined.

$$\begin{gathered} \left(\text{a}\right)\frac{2x^2}{z} \\ \left(\text{b}\right)\frac{4p-1}{6p-5} \\ \left(\text{c}\right)\frac{n-3}{n^2+2n-8} \end{gathered}$$

In the following exercises, determine the values for which the rational expression is defined.

$$\begin{gathered} \left(\text{a}\right)\frac{10m}{11n} \\ \left(\text{b}\right)\frac{6y+13}{4y-9} \\ \left(\text{c}\right)\frac{b-8}{b^2-36} \end{gathered}$$

In the following exercises, determine the values for which the rational expression is defined.

$$\begin{gathered} \left(\text{a}\right)\frac{4x^{2}y}{3y} \\ \left(\text{b}\right)\frac{3x-2}{2x+1} \\ \left(\text{c}\right)\frac{u-1}{u^2-3u-28} \end{gathered}$$

In the following exercises, determine the values for which the rational expression is defined.

$$\begin{gathered} \left(\text{a}\right)\frac{5p{q}^2}{9q} \\ \left(\text{b}\right)\frac{7a-4}{3a+5} \\ \left(\text{c}\right)\frac{1}{x^2-4} \end{gathered}$$

In the following exercises, simplify each rational expression.

$-\frac{44}{55}$

In the following exercises, simplify each rational expression.

$\frac{56}{63}$

Simplify the rational expression $\frac{36v^3{w}^2}{27v{w}^3} .$

Simplify the rational expression $\frac{8n-96}{3n-36}$.

Simplify the rational expression $\frac{12p-240}{5p-100} .$

Simplify the rational expression $\frac{y^2+3y-4}{y^2-6y+5}$

Simplify the rational expression $\frac{a^2-4}{a^2+6y-16}$

Simplify the rational expression $\frac{y^2-2y-3}{y^2-9} .$

Simplify the rational expression  $\frac{x^3-2x^2-25x+50}{x^2-25}$

Simplify the rational expression$\frac{8b^2-32}{2b^2-6b-80}$

Simplify the rational expression $\frac{-5c^2-10c}{-10c^2+30c+100}.$

Simplify the rational expression $\frac{5r^2+30rs-35s^2}{r^2-49s^2}$$\frac{5r^2+30rs-35s^2}{r^2-49s^2}$

simplify the rational expression $\frac{(a-5)}{(5-a)}$

Simplify the rational expression $\frac{5-d}{d-5}$

Simplify the rational expression $\frac{4v-32}{64-v^2}$

Simplify the rational expression $\frac{w^3+216}{w^2-36}$

Simplify the rational expression $\frac{v^3+125}{v^2-25}$

Simplify the rational expression $\frac{z^2-9z+20}{16-z^2}$

Simplify rational expression is $\frac{z^2-9z+20}{16-z^2}$

Simplify the rational expression $\frac{a^2-5z-36}{81-a^2}$