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Section 1
Multiplying and Dividing Rational Expressions
Simplify each expression.$\frac{45 m n^{3}}{20 n^{7}}$
Simplify each expression.$\frac{a+b}{a^{2}-b^{2}}$
Simplify each expression.$\frac{x^{2}+6 x+9}{x+3}$
Simplify each expression.$\frac{36 c^{3} d^{2}}{54 c d^{5}}$
Identify all values of $y$ for which $\frac{y-4}{y^{2}-4 y-12}$ is undefined.A. -2, 4, 6 B. -6, -4, 2 C. -2, 0, 6 D. -2, 6
Simplify each expression.$\frac{9 y^{2}-6 y^{3}}{2 y^{2}+5 y-12}$
Simplify each expression.$\frac{b^{3}-a^{3}}{a^{2}-b^{2}}$
Simplify each expression.$\frac{2 a^{2}}{5 b^{2} c} \cdot \frac{3 b c^{3}}{8 a^{2}}$
Simplify each expression.$\frac{3 t+6}{7 t-7} \cdot \frac{14 t-14}{5 t+10}$
Simplify each expression.$\frac{35}{16 x^{2}} \div \frac{21}{4 x}$
Simplify each expression.$\frac{20 x y^{3}}{21} \div \frac{15 x^{3} y^{2}}{14}$
Simplify each expression.$\frac{12 p^{2}+6 p-6}{4(p+1)^{2}} \div \frac{6 p-3}{2 p+10}$
Simplify each expression.$\frac{x^{2}+6 x+9}{x^{2}+7 x+6} \div \frac{4 x+12}{3 x+3}$
Simplify each expression.$\frac{\frac{c^{3} d^{3}}{a}}{\frac{x c^{2} d}{a x^{2}}}$
Simplify each expression.$\frac{\frac{2 y}{y^{2}-4}}{\frac{3}{y^{2}-4 y+4}}$
Simplify each expression.$\frac{30 b c}{12 b^{3}}$
Simplify each expression.$\frac{-3 m n^{3}}{21 m^{2} n^{2}}$
Simplify each expression.$\frac{5 t-5}{t^{2}-1}$
Simplify each expression.$\frac{c+5}{2 c+10}$
Simplify each expression.$\frac{3 t-6}{2-t}$
Simplify each expression.$\frac{9-t^{2}}{t^{2}+t-12}$
Simplify each expression.$\frac{3 x y z}{4 x z} \cdot \frac{6 x^{2}}{3 y^{2}}$
Simplify each expression.$\frac{-4 a b}{21 c} \cdot \frac{14 c^{2}}{18 a^{2}}$
Simplify each expression.$\frac{3}{5 d} \div\left(-\frac{9}{15 d f}\right)$
Simplify each expression.$\frac{p^{3}}{2 q} \div \frac{-p}{4 q}$
Simplify each expression.$\frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}}$
Simplify each expression.$\frac{4 w+4}{3} \cdot \frac{1}{w+1}$
Simplify each expression.$\frac{4 t^{2}-4}{9(t+1)^{2}} \cdot \frac{3 t+3}{2 t-2}$
Simplify each expression.$\frac{3 p-21}{p^{2}-49} \cdot \frac{p^{2}-7 p}{3 p}$
Simplify each expression.$\frac{\frac{m^{3}}{3 n}}{-\frac{m^{4}}{9 n^{2}}}$
Simplify each expression.$\frac{\frac{p^{3}}{2 q}}{-\frac{p^{2}}{4 q}}$
Simplify each expression.$\frac{\frac{m+n}{5}}{\frac{m^{2}+n^{2}}{5}}$
Simplify each expression.$\frac{\frac{x+y}{2 x-y}}{\frac{x+y}{2 x+y}}$
Simplify each expression.Under what conditions is $\frac{x-4}{(x+5)(x-1)}$ undefined?
Simplify each expression.For what values is $\frac{2 d(d+1)}{(d+1)\left(d^{2}-4\right)}$ undefined?
A parallelogram with an area of $6 x^{2}-7 x-5$ square units has a base of $3 x-5$ units. Determine the height of the parallelogram.
GEOMETRY Parallelogram $F$ has an area of $8 x^{2}+10 x-3$ square meters and a height of $2 x+3$ meters. Parallelogram $G$ has an area of $6 x^{2}+13 x-5$ square meters and a height of $3 x-1$ meters. Find the area of right triangle $H .$
Simplify each expression.$\frac{\left(-3 x^{2} y\right)^{3}}{9 x^{2} y^{2}}$
Simplify each expression.$\frac{\left(-2 r s^{2}\right)^{2}}{12 r^{2} s^{3}}$
Simplify each expression.$\frac{\left(-5 m n^{2}\right)^{3}}{5 m^{2} n^{4}}$
Simplify each expression.$\frac{y^{2}+4 y+4}{3 y^{2}+5 y-2}$
Simplify each expression.$\frac{a^{2}+2 a+1}{2 a^{2}+3 a+1}$
Simplify each expression.$\frac{3 x^{2}-2 x-8}{3 x^{2}-12}$
Simplify each expression.$\frac{a^{2}-4}{6-3 a}$
Simplify each expression.$\frac{b^{2}-4 b+3}{3-2 b-b^{2}}$
Simplify each expression.$\frac{6 x^{2}-6}{14 x^{2}-28 x+14}$
Simplify each expression.$\frac{25 a^{2} b^{3}}{6 x^{2} y} \cdot \frac{8 x y^{2}}{20 a^{3} b^{2}}$
Simplify each expression.$\frac{-9 c d}{8 x w} \cdot \frac{(-4 w)^{2}}{15 c}$
Simplify each expression.$\frac{2 x^{3} y}{z^{5}} \div\left(\frac{4 x y}{z^{3}}\right)^{2}$
Simplify each expression.$\frac{w^{2}-11 w+24}{w^{2}-18 w+80} \cdot \frac{w^{2}-15 w+50}{w^{2}-9 w+20}$
Simplify each expression.$\frac{r^{2}+2 r-8}{r^{2}+4 r+3} \div \frac{r-2}{3 r+3}$
Simplify each expression.$\frac{\frac{5 x^{2}-5 x-30}{45-15 x}}{\frac{6+x-x^{2}}{4 x-12}}$
Under what conditions is $\frac{a^{2}+a b+b^{2}}{a^{2}-b^{2}}$ undefined?
At the end of the 2005-2006 season, the Seattle Sonics’ Ray Allen had made 5422 field goals out of 12,138 attempts during his NBA career.Write a ratio to represent the ratio of the number of career field goals made to career field goals attempted by Ray Allen at the end of the 2005-2006 season.
At the end of the 2005-2006 season, the Seattle Sonics’ Ray Allen had made 5422 field goals out of 12,138 attempts during his NBA career.Suppose Ray Allen attempted $a$ field goals and made $m$ field goals during the 2006-2007 season. Write a rational expression to represent the ratio of the number of career field goals made to the number of career field goals attempted at the end of the 2006-2007 season.
An airplane is traveling at the rate $r$ of 500 miles per hour for a time $t$ of $(6+x)$ hours. A second airplane travels at the rate $r$ of $(540+90 x)$ miles per hour for a time $t$ of 6 hours.
Write a rational expression to represent the ratio of the distance $d$ traveled by the first airplane to the distance $d$ traveled by the second airplane.
Simplify the rational expression. What does this expression tell you about the distances traveled of the two airplanes?
Under what condition is the rational expression undefined? Describe what this condition would tell you about the two airplanes.
Consider $f(x)=\frac{-15 x^{2}+10 x}{5 x}$ and $g(x)=-3 x+2$.Simplify $\frac{-15 x^{2}+10 x}{5 x} .$ What do you observe about the expression?
Consider $f(x)=\frac{-15 x^{2}+10 x}{5 x}$ and $g(x)=-3 x+2$.Graph $f(x)$ and $g(x)$ on a graphing calculator. How do the graphs appear?
Consider $f(x)=\frac{-15 x^{2}+10 x}{5 x}$ and $g(x)=-3 x+2$.Use the table feature to examine the function values for $f(x)$ and $g(x) .$ How do the tables compare?
Consider $f(x)=\frac{-15 x^{2}+10 x}{5 x}$ and $g(x)=-3 x+2$.How can you use what you have observed with $f(x)$ and $g(x)$ to verify that expressions are equivalent or to identify excluded values?
Write two rational expressions that are equivalent.
Rewrite $\frac{a+\sqrt{b}}{-a^{2}+b}$ so it has a numerator of 1.
Identify the expression that does not belong with the other three. Explain your reasoning.
Determine whether $\frac{2 d+5}{3 d+5}=\frac{2}{3}$ is sometimes, always, or never true. Explain.
Use the information about rational expressions on page 442 to explain how rational expressions are used in mixtures. Include an example of a mixture problem that could be represented by $\frac{8+x}{13+x+y}$.
For what value(s) of $x$ is $\frac{4 x}{x^{2}-x}$ undefined?A -1, 1B -1, 0, 1C 0, 1D 0
Which is the simplified form of $\frac{4 x^{3} y^{2} z^{-1}}{\left(x^{-2} y^{3} z^{2}\right)^{2}} ?$F. $\frac{4 x^{7}}{y^{4} z^{5}}$G. $\frac{4 x y}{z^{5}}$H. $\frac{4}{y^{3} z^{5}}$J. $\frac{4}{x y^{4} z^{5}}$
Graph each function. State the domain and range.$y=\sqrt{x-2}$
Graph each function. State the domain and range.$y=\sqrt{x}-1$
Graph each function. State the domain and range.$y=2 \sqrt{x}+1$
Determine whether $f(x)=x-2$ and $g(x)=2 x$ are inverse functions.
Determine whether each graph represents an odd-degree or an even-degree polynomial function. Then state how many real zeros each function has.
Earth is an average of $1.496 \times 10^{8}$ kilometers from the Sun. If light travels $3 \times 10^{5}$ kilometers per second, how long does it take sunlight to reach Earth?
Solve each equation by factoring.$r^{2}-3 r=4$
Solve each equation by factoring.$18 u^{2}-3 u=1$
Solve each equation by factoring.$d^{2}-5 d=0$
Solve each equation.$|2 x+7|+5=0$
Solve each equation.$5|3 x-4|=x+1$
Solve each equation.$\frac{2}{3}+x=-\frac{4}{9}$
Solve each equation.$x+\frac{5}{8}=-\frac{5}{6}$
Solve each equation.$x-\frac{3}{5}=\frac{2}{3}$
Solve each equation.$x+\frac{3}{16}=-\frac{1}{2}$
Solve each equation.$x-\frac{1}{6}=-\frac{7}{9}$
Solve each equation.$x-\frac{3}{8}=-\frac{5}{24}$