5.4E: Exercises

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Practice Makes Perfect

Undefined Values

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

\(\dfrac{2x^2}{z}\)
\(\dfrac{4p−1}{6p−5}\)
\(\dfrac{n−3}{n^2+2n−8}\)
\(\dfrac{4x^2y}{3y}\)
\(\dfrac{3x−2}{2x+1}\)
\(\dfrac{u−1}{u^2−3u−28}\)

Answer

\(z=0\)
\(p=\dfrac{5}{6}\)
\(n=−4, n=2\)
\(y=0\)
\(x=−\dfrac{1}{2}\)
\(u=−4, u=7\)

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

\(\dfrac{x+1+y}{y+x+1}\)
\(\dfrac{8m^3n}{12mn^2}\)
\(\dfrac{8n−96}{3n−36}\)
\(\dfrac{x^2+4x−5}{x^2−2x+1}\)
\(\dfrac{a^2−4}{a^2+6a−16}\)
\(\dfrac{p^3+3p^2+4p+12}{p^2+p−6}\)
\(\dfrac{8b^2−32b}{2b^2−6b−80}\)
\(\dfrac{3m^2+30mn+75n^2}{4m^2−100n^2}\)
\(\dfrac{a−5}{5−a}\)
\(\dfrac{20−5y}{y^2−16}\)
\(\dfrac{z^2−9z+20}{16−z^2}\)

Answer

1
\(\dfrac{2m^2}{3n}\)
\(\dfrac{8}{3}\)
\(\dfrac{x+5}{x−1}\)
\(\dfrac{a+2}{a+8}\)
\(\dfrac{p^2+4}{p−2}\)
\(\dfrac{4b(b−4)}{(b+5)(b−8)}\)
\(\dfrac{3(m+5n)}{4(m−5n)}\)
\(−1\)
\(\dfrac{−5}{y+4}\)
\(-\dfrac{z-5}{4+z}\)

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

\(\dfrac{12}{16}\cdot \dfrac{4}{10}\)
\(\dfrac{5x^2y^4}{12xy^3}\cdot \dfrac{6x^2}{20y^2}\)
\(\dfrac{5p^2}{p^2−5p−36}\cdot \dfrac{p^2−16}{10p}\)
\(\dfrac{2y^2−10y}{y^2+10y+25}\cdot \dfrac{y+5}{6y}\)
\(\dfrac{28−4b}{3b−3}\cdot \dfrac{b^2+8b−9}{b^2−49}\)
\(\dfrac{c^2-10c+25}{c^2−25}\cdot \dfrac{c^2+10c+25}{3c^2−14c−5}\)
\(\dfrac{2m^2−3m−2}{2m^2+7m+3}\cdot \dfrac{3m^2−14m+15}{3m^2+17m−20}\)

Answer

\(\dfrac{3}{10}\)
\(\dfrac{x^3}{8y}\)
\(\dfrac{p(p−4)}{2(p−9)}\)
\(\dfrac{y−5}{3(y+5)}\)
\(\dfrac{−4(b+9)}{3(b+7)}\)
\(\dfrac{c+5}{3c+1}\)
\(\dfrac{(m−3)(m−2)(3m-5)}{(m-1)(m+3)(3m+20)}\)

Divide Rational Expressions

In the following exercises, divide the rational expressions.

\(\dfrac{v−5}{11−v}÷\dfrac{v^2−25}{v−11}\)
\(\dfrac{3s^2}{s^2−16}÷\dfrac{s^3+4s^2+16s}{s^3−64}\)
\(\dfrac{p^3+q^3}{3p^2+3pq+3q^2}÷\dfrac{p^2−q^2}{12}\)
\(\dfrac{x^2+3x−10}{4x}÷(2x^2+20x+50)\)
\(\dfrac{\dfrac{2a^2−a−21}{5a+20}}{\dfrac{a^2+7a+12}{a^2+8a+16}}\)
\(\dfrac{\dfrac{12c^2−12}{2c^2−3c+1}}{\dfrac{4c+4}{6c^2−13c+5}}\)
\(\dfrac{10m^2+80m}{3m−9}·\dfrac{m^2+4m−21}{m^2−9m+20}÷\dfrac{5m^2+10m}{2m−10}\)
\(\dfrac{12p^2+3p}{p+3}÷\dfrac{p^2+2p−63}{p^2−p−12}·\dfrac{p−7}{9p^3−9p^2}\)

Answer

\(−\dfrac{1}{v+5}\)
\(\dfrac{3s}{s+4}\)
\(\dfrac{4(p^2−pq+q^2)}{(p−q)(p^2+pq+q^2)}\)
\(\dfrac{x−2}{8x(x+5)}\)
\(\dfrac{2a−7}{5}\)
\(3(3c−5)\)
\(\dfrac{4(m+8)(m+7)}{3(m−4)(m+2)}\)
\(\dfrac{(4p+1)(p−4)}{3p(p+9)(p−1)}\)

Writing Exercises

33. Explain how you find the values of x for which the rational expression \(\dfrac{x^2−x−20}{x^2−4}\) is undefined.

34.

a. Multiply \(\dfrac{7}{4}·\dfrac{9}{10}\) and explain all your steps.

b. Multiply \(\dfrac{n}{n−3}·\dfrac{9}{n+3}\) 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 a.? Why or why not?

Answer: Answers will vary.