8.2 Multiply and Divide Rational Expressions

If you miss a problem, go back to the section listed and review the material.

Multiply: $\frac{14}{15} \cdot \frac{6}{35} .$
If you missed this problem, review the reference.

Divide: $\frac{14}{15} \div \frac{6}{35} .$
If you missed this problem, review the reference.

Factor completely: $2 x^{2} - 98 .$
If you missed this problem, review the reference.

Factor completely: $10 n^{3} + 10 .$
If you missed this problem, review the reference.

Factor completely: $10 p^{2} - 25 p q - 15 q^{2} .$
If you missed this problem, review the reference.

Mulitply: $\frac{6}{10} \cdot \frac{15}{12} .$

Mulitply: $\frac{20}{15} \cdot \frac{6}{8} .$

Mulitply: $\frac{3 p q}{q^{2}} \cdot \frac{5 p^{2} q}{6 p q} .$

Mulitply: $\frac{6 x^{3} y}{7 x^{2}} \cdot \frac{2 x y^{3}}{x^{2} y} .$

Mulitply: $\frac{5 x}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 4}{10 x} .$

Mulitply: $\frac{9 x^{2}}{x^{2} + 11 x + 30} \cdot \frac{x^{2} - 36}{3 x^{2}} .$

Multiply: $\frac{n^{2} - 7 n}{n^{2} + 2 n + 1} \cdot \frac{n + 1}{2 n} .$

Multiply: $\frac{x^{2} - 25}{x^{2} - 3 x - 10} \cdot \frac{x + 2}{x} .$

Multiply: $\frac{x^{2} - 4 x}{x^{2} + 5 x + 6} \cdot \frac{x + 2}{x} .$

Multiply: $\frac{16 - 4 x}{2 x - 12} \cdot \frac{x^{2} - 5 x - 6}{x^{2} - 16} .$

Multiply: $\frac{12 x - 6 x^{2}}{x^{2} + 8 x} \cdot \frac{x^{2} + 11 x + 24}{x^{2} - 4} .$

Multiply: $\frac{9 v - 3 v^{2}}{9 v + 36} \cdot \frac{v^{2} + 7 v + 12}{v^{2} - 9} .$

Multiply: $\frac{3 a - 21}{a^{2} - 9 a + 14} \cdot \frac{a^{2} - 4}{3 a + 6} .$

Multiply: $\frac{b^{2} - b}{b^{2} + 9 b - 10} \cdot \frac{b^{2} - 100}{b^{2} - 10 b} .$

Divide: $\frac{c + 3}{5 - c} \div \frac{c^{2} - 9}{c - 5} .$

Divide: $\frac{2 - d}{d - 4} \div \frac{4 - d^{2}}{4 - d} .$

Divide: $\frac{2 m^{2}}{m^{2} - 8 m} \div \frac{8 m^{2} + 24 m}{m^{2} + m - 6} .$

Divide: $\frac{15 n^{2}}{3 n^{2} + 33 n} \div \frac{5 n - 5}{n^{2} + 9 n - 22} .$

Divide: $\frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \div \frac{2 x^{2} - 13 x + 15}{x^{2} - 8 x + 16} .$

Divide: $\frac{3 a^{2} - 8 a - 3}{a^{2} - 25} \div \frac{3 a^{2} - 14 a - 5}{a^{2} + 10 a + 25} .$

Divide: $\frac{4 b^{2} + 7 b - 2}{1 - b^{2}} \div \frac{4 b^{2} + 15 b - 4}{b^{2} - 2 b + 1} .$

Divide: $\frac{p^{3} + q^{3}}{2 p^{2} + 2 p q + 2 q^{2}} \div \frac{p^{2} - q^{2}}{6} .$

Divide: $\frac{x^{3} - 8}{3 x^{2} - 6 x + 12} \div \frac{x^{2} - 4}{6} .$

Divide: $\frac{2 z^{2}}{z^{2} - 1} \div \frac{z^{3} - z^{2} + z}{z^{3} - 1} .$

Divide: $\frac{a^{2} - b^{2}}{3 a b} \div \left(a^{2} + 2 a b + b^{2}\right) .$

Divide: $\frac{2 x^{2} - 14 x - 16}{4} \div \left(x^{2} + 2 x + 1\right) .$

Divide: $\frac{y^{2} - 6 y + 8}{y^{2} - 4 y} \div \left(3 y^{2} - 12 y\right) .$

Divide: $\frac{\frac{6 x^{2} - 7 x + 2}{4 x - 8}}{\frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x + 6}} .$

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

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

Divide: $\frac{4 m + 4}{3 m - 15} \cdot \frac{m^{2} - 3 m - 10}{m^{2} - 4 m - 32} \div \frac{12 m - 36}{6 m - 48} .$

Divide: $\frac{2 n^{2} + 10 n}{n - 1} \div \frac{n^{2} + 10 n + 24}{n^{2} + 8 n - 9} \cdot \frac{n + 4}{8 n^{2} + 12 n} .$

Multiply.

$\frac{12}{16} \cdot \frac{4}{10}$

Multiply.

$\frac{18}{10} \cdot \frac{4}{30}$

Multiply.

$\frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}}$

Multiply.

$\frac{12 a^{3} b}{b^{2}} \cdot \frac{2 a b^{2}}{9 b^{3}}$

Multiply.

$\frac{5 p^{2}}{p^{2} - 5 p - 36} \cdot \frac{p^{2} - 16}{10 p}$

Multiply.

$\frac{4 r}{r^{2} - 3 r - 10} \cdot \frac{r^{2} - 25}{8 r^{2}}$

Multiply.

$\frac{x^{2} - 7 x}{x^{2} + 6 x + 9} \cdot \frac{x + 3}{4 x}$

Multiply.

$\frac{z^{2} + 3 z}{z^{2} - 3 z - 4} \cdot \frac{z - 4}{z^{2}}$

Multiply.

$\frac{28 - 4 b}{3 b - 3} \cdot \frac{b^{2} + 8 b - 9}{b^{2} - 49}$

Multiply.

$\frac{35 d - 7 d^{2}}{d^{2} + 7 d} \cdot \frac{d^{2} + 12 d + 35}{d^{2} - 25}$

Multiply.

$\frac{4 n + 20}{n^{2} + n - 20} \cdot \frac{n^{2} - 16}{4 n + 16}$

Multiply.

$\frac{q^{2} - 2 q}{q^{2} + 6 q - 16} \cdot \frac{q^{2} - 64}{q^{2} - 8 q}$

Divide.

$\frac{t - 6}{3 - t} \div \frac{t - 5}{t^{2} - 9}$

Divide.

$\frac{10 + w}{w - 8} \div \frac{100 - w^{2}}{8 - w}$

Divide.

$\frac{27 y^{2}}{3 y - 21} \div \frac{3 y^{2} + 18}{y^{2} + 13 y + 42}$

Divide.

$\frac{16 a^{2}}{4 a + 36} \div \frac{4 a^{2} - 24 a}{a^{2} + 4 a - 45}$

Divide.

$\frac{5 c^{2} + 9 c - 2}{c^{2} - 4} \div \frac{5 c^{2} - 16 c + 3}{c^{2} + 4 c + 4}$

Divide.

$\frac{6 m^{2} - 11 m - 2}{9 - m^{2}} \div \frac{6 m^{2} + 25 m + 4}{m^{2} - 6 m + 9}$

Divide.

$\frac{3 s^{2}}{s^{2} - 16} \div \frac{s^{3} + 4 s^{2} + 16 s}{s^{3} - 64}$

Divide.

$\frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12}$

Divide.

$\frac{t^{2} - 9}{2 t} \div \left(t^{2} - 6 t + 9\right)$

Divide.

$\frac{2 y^{2} - 10 y z - 48 z^{2}}{2 y - 1} \div \left(4 y^{2} - 32 y z\right)$

Divide.

$\frac{\frac{2 a^{2} - a - 21}{5 a + 20}}{\frac{a^{2} + 7 a + 12}{a^{2} + 8 a + 16}}$

Divide.

$\frac{\frac{12 c^{2} - 12}{2 c^{2} - 3 c + 1}}{\frac{4 c + 4}{6 c^{2} - 13 c + 5}}$

Divide.

$\frac{10 m^{2} + 80 m}{3 m - 9} \cdot \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20}$
$\div \frac{5 m^{2} + 10 m}{2 m - 10}$

Divide.

$\frac{12 p^{2} + 3 p}{p + 3} \div \frac{p^{2} + 2 p - 63}{p^{2} - p - 12}$
$\cdot \frac{p - 7}{9 p^{3} - 9 p^{2}}$

Probability The director of large company is interviewing applicants for two identical jobs. If $w =$ the number of women applicants and $m =$ the number of men applicants, then the probability that two women are selected for the jobs is $\frac{w}{w + m} \cdot \frac{w - 1}{w + m - 1} .$

1. ⓐ Simplify the probability by multiplying the two rational expressions.
2. ⓑ Find the probability that two women are selected when $w = 5$ and $m = 10$.

1. ⓐ Multiply $\frac{7}{4} \cdot \frac{9}{10}$ and explain all your steps.
2. ⓑ Multiply $\frac{n}{n - 3} \cdot \frac{9}{n + 3}$ and explain all your steps.
3. ⓒ Evaluate your answer to part (b) when $n = 7 .$ Did you get the same answer you got in part (a)? Why or why not?