Section 1
Multiply and Divide Rational Expressions
In the following exercises, determine the values for which the rational expression is undefined.(a) $\frac{2 x^2}{z}$(6) $\frac{4 p-1}{6 p-5}$(c) $\frac{n-3}{n^2+2 n-8}$
In the following exercises, determine the values for which the rational expression is undefined.
(a) $\frac{10 m}{11 n}$(b) $\frac{6 y+13}{4 y-9}$(c) $\frac{b-8}{b^2-36}$
(a) $\frac{4 x^2 y}{3 y}$(b) $\frac{3 x-2}{2 x+1}$(c) $\frac{u-1}{u^2-3 u-28}$
(a) $\frac{5 p q^2}{9 q}$(b) $\frac{7 a-4}{3 a+5}$(c) $\frac{1}{x^2-4}$
In the following exercises, simplify each rational expression. $-\frac{44}{55}$
In the following exercises, simplify each rational expression.$\frac{56}{63}$
In the following exercises, simplify each rational expression. $\frac{8 m^3 n}{12 m n^2}$
In the following exercises, simplify each rational expression. $\frac{36 v^3 w^2}{27 v w^3}$
In the following exercises, simplify each rational expression. $\frac{8 n-96}{3 n-36}$
In the following exercises, simplify each rational expression.$\frac{12 p-240}{5 p-100}$
In the following exercises, simplify each rational expression.$\frac{x^2+4 x-5}{x^2-2 x+1}$
In the following exercises, simplify each rational expression.$\frac{y^2+3 y-4}{y^2-6 y+5}$
In the following exercises, simplify each rational expression.$\frac{a^2-4}{a^2+6 a-16}$
In the following exercises, simplify each rational expression.$\frac{y^2-2 y-3}{y^2-9}$
In the following exercises, simplify each rational expression.$\frac{p^3+3 p^2+4 p+12}{p^2+p-6}$
In the following exercises, simplify each rational expression.$\frac{x^3-2 x^2-25 x+50}{x^2-25}$
In the following exercises, simplify each rational expression.$\frac{8 b^2-32 b}{2 b^2-6 b-80}$
In the following exercises, simplify each rational expression.$\frac{-5 c^2-10 c}{-10 c^2+30 c+100}$
In the following exercises, simplify each rational expression.$\frac{3 m^2+30 m n+75 n^2}{4 m^2-100 n^2}$
In the following exercises, simplify each rational expression.$\frac{5 r^2+30 r s-35 s^2}{r^2-49 s^2}$
In the following exercises, simplify each rational expression.$\frac{a-5}{5-a}$
In the following exercises, simplify each rational expression.$\frac{5-d}{d-5}$
In the following exercises, simplify each rational expression.$\frac{20-5 y}{y^2-16}$
In the following exercises, simplify each rational expression.$\frac{4 y-32}{64-y^2}$
In the following exercises, simplify each rational expression.$\frac{w^3+216}{w^2-36}$
In the following exercises, simplify each rational expression.$\frac{v^3+125}{v^2-25}$
In the following exercises, simplify each rational expression.$\frac{z^2-9 z+20}{16-z^2}$
In the following exercises, simplify each rational expression.$\frac{a^2-5 z-36}{81-a^2}$
In the following exercises, multiply the rational expressions.$\frac{12}{16} \cdot \frac{4}{10}$
In the following exercises, multiply the rational expressions.$\frac{32}{5} \cdot \frac{16}{24}$
In the following exercises, multiply the rational expressions.$\frac{5 x^2 y^4}{12 x y^3} \cdot \frac{6 x^2}{20 y^2}$
In the following exercises, multiply the rational expressions.$\frac{12 a^3 b}{b^2} \cdot \frac{2 a b^2}{9 b^3}$
In the following exercises, multiply the rational expressions.$\frac{5 p^2}{p^2-5 p-36} \cdot \frac{p^2-16}{10 p}$
In the following exercises, multiply the rational expressions.$\frac{3 q^2}{q^2+q-6} \cdot \frac{q^2-9}{9 q}$
In the following exercises, multiply the rational expressions.$\frac{2 y^2-10 y}{y^2+10 y+25} \cdot \frac{y+5}{6 y}$
In the following exercises, multiply the rational expressions.$\frac{z^2+3 z}{z^2-3 z-4} \cdot \frac{z-4}{z^2}$
In the following exercises, multiply the rational expressions.$\frac{28-4 b}{3 b-3} \cdot \frac{b^2+8 b-9}{b^2-49}$
In the following exercises, multiply the rational expressions.$\frac{72 m-12 m^2}{8 m+32} \cdot \frac{m^2+10 m+24}{m^2-36}$
In the following exercises, multiply the rational expressions.$\frac{5 c^2+9 c+2}{c^2-25} \cdot \frac{c^2+10 c+25}{3 c^2-14 c-5}$
In the following exercises, multiply the rational expressions.$\frac{2 d^2+d-3}{d^2-16} \cdot \frac{d^2-8 d+16}{2 d^2-9 d-18}$
In the following exercises, multiply the rational expressions.$\frac{6 m^2-2 m-10}{9-m^2} \cdot \frac{m^2-6 m+9}{6 m^2+29 m-20}$
In the following exercises, multiply the rational expressions. $\frac{2 n^2-3 n-14}{25-n^2} \cdot \frac{n^2-10 n+25}{2 n^2-13 n+21}$
In the following exercises, divide the rational expressions.$\frac{v-5}{11-v} \div \frac{v^2-25}{v-11}$
In the following exercises, divide the rational expressions. $\frac{10+w}{w-8} \div \frac{100-w^2}{8-w}$
In the following exercises, divide the rational expressions. $\frac{3 s^2}{s^2-16} \div \frac{s^3-4 s^2+16 s}{s^3-64}$
In the following exercises, divide the rational expressions.$\frac{r^2-9}{15} \div \frac{r^3-27}{5 r^2+15 r+45}$
In the following exercises, divide the rational expressions.$\frac{p^3+q^3}{3 p^2+3 p q+3 q^2} \div \frac{p^2-q^2}{12}$
In the following exercises, divide the rational expressions.$\frac{v^3-8 w^3}{2 v^2+4 v w+8 w^2} \div \frac{v^2-4 w^2}{4}$
In the following exercises, divide the rational expressions.$\frac{x^2+3 x-10}{4 x} \div\left(2 x^2+20 x+50\right)$
In the following exercises, divide the rational expressions.$\frac{2 y^2-10 y z-48 z^2}{2 y-1} \div\left(4 y^2-32 y z\right)$
In the following exercises, divide the rational expressions.$\frac{\frac{2 a^2-a-21}{5 a+70}}{\frac{a^2+7 a+12}{a^2+8 a+16}}$
In the following exercises, divide the rational expressions.$\frac{\frac{3 b^2+2 b-8}{12 b+18}}{\frac{3 b^2+2 b-8}{2 b^2-7 b-15}}$
In the following exercises, divide the rational expressions.$\frac{\frac{12 c^2-12}{2 c^2-3 c+1}}{\frac{4 c+4}{6 c^2-13 c+5}}$
In the following exercises, divide the rational expressions.$\frac{\frac{4 d^2+7 d-2}{35 d+10}}{\frac{d^2-4}{7 d^2-12 d-4}}$
For the following exercises, perform the indicated operations.$\frac{10 m^2+80 m}{3 m-9} \cdot \frac{m^2+4 m-21}{m^2-9 m+20} \div \frac{5 m^2+10 m}{2 m-10}$
For the following exercises, perform the indicated operations.$\frac{4 n^2+32 n}{3 n+2} \cdot \frac{3 n^2-n-2}{n^2+n-30} \div \frac{108 n^2-24 n}{n+6}$
For the following exercises, perform the indicated operations.$\frac{12 p^2+3 p}{p+3} \div \frac{p^2+2 p-63}{p^2-p-12} \cdot \frac{p-7}{9 p^3-9 p^2}$
For the following exercises, perform the indicated operations.$\frac{6 q+3}{9 q^2-9 q} \div \frac{q^2+14 q+33}{q^2+4 q-5} \cdot \frac{4 q^2+12 q}{12 q+6}$
In the following exercises, find the domain of each function.$R(x)=\frac{x^3-2 x^2-25 x+50}{x^2-25}$
In the following exercises, find the domain of each function.$R(x)=\frac{x^3+3 x^2-4 x-12}{x^2-4}$
In the following exercises, find the domain of each function.$R(x)=\frac{3 x^2+15 x}{6 x^2+6 x-36}$
In the following exercises, find the domain of each function.$R(x)=\frac{8 x^2-32 x}{2 x^2-6 x-80}$
For the following exercises, find $R(x)=f(x) \cdot g(x)$ where $f(x)$ and $g(x)$ are given.
$$\begin{aligned}& f(x)=\frac{6 x^2-12 x}{x^2+7 x-18} \\& g(x)=\frac{x^2-81}{3 x^2-27 x}\end{aligned}$$
$$\begin{aligned}& f(x)=\frac{x^2-2 x}{x^2+6 x-16} \\& g(x)=\frac{x^2-64}{x^2-8 x}\end{aligned}$$
$$\begin{aligned}& f(x)=\frac{4 x}{x^2-3 x-10} \\& g(x)=\frac{x^2-25}{8 x^2}\end{aligned}$$
$$\begin{aligned}& f(x)=\frac{2 x^2+8 x}{x^2-9 x+20} \\& g(x)=\frac{x-5}{x^2}\end{aligned}$$
For the following exercises, find $R(x)=\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are given.
$$\begin{aligned}& f(x)=\frac{27 x^2}{3 x-21} \\& g(x)=\frac{3 x^2+18 x}{x^2+13 x+42}\end{aligned}$$
$$\begin{aligned}& f(x)=\frac{24 x^2}{2 x-8} \\& g(x)=\frac{4 x^3+28 x^2}{x^2+11 x+28}\end{aligned}$$
$$\begin{aligned}& f(x)=\frac{16 x^2}{4 x+36} \\& g(x)=\frac{4 x^2-24 x}{x^2+4 x-45}\end{aligned}$$
$$\begin{aligned}& f(x)=\frac{24 x^2}{2 x-4} \\& g(x)=\frac{12 x^2+36 x}{x^2-11 x+18}\end{aligned}$$
Explain how you find the values of $x$ for which the rational expression $\frac{x^2-x-20}{x^2-4}$ is undefined.
Explain all the steps you take to simplify the rational expression $\frac{p^2+4 p-21}{9-p^2}$.
Multiply $\frac{7}{4}, \frac{9}{10}$ and explain all your steps.(b) Multiply $\frac{n}{n-3} \cdot \frac{9}{n+3}$ and explain all your steps.(c) Evaluate your answer to part (b) when $n=7$. Did you get the same answer you got in part (3)? Why or why not?
(a) Divide $\frac{24}{5} \div 6$ and explain all your steps.(b) Divide $\frac{x^2-1}{x} \div(x+1)$ and explain all your steps.(c) Evaluate your answer to part (6) when $x=5$. Did you get the same answer you got in part (3)? Why or why not?