Chapter 6

Simplify each complex fraction. $$ \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} $$

Add or subtract, and then simplify, if possible. See Example 7. $$\frac{8 x}{x-4}-\frac{10 x}{4-x}$$

Solve each proportion. $$ \frac{b^{2}}{5}=\frac{b}{6 b-13} $$

Perform each division. \(\left(x^{3}+3 x+5 x^{2}+6+x^{4}\right) \div\left(x^{2}+3\right)\)

Divide, and then simplify, if possible. See Example 7. $$ \frac{8 y^{2}-14 y-15}{6 y^{2}-11 y-10} \div \frac{4 y^{2}-9 y-9}{3 y^{2}-7 y-6} $$

Simplify each rational expression. $$ \frac{5 s^{2}-4 s t-t^{2}}{s t-s^{2}} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{-10}{t+3}=1-\frac{11}{t-3}\)

Simplify each complex fraction. $$ \frac{\frac{1}{x^{2}}-\frac{3}{x y}+\frac{2}{y^{2}}}{\frac{2}{x^{2}}-\frac{1}{x y}-\frac{1}{y^{2}}} $$

Solve each proportion. $$ \frac{x}{x+2}=\frac{6}{x+2} $$

Perform each division. \(\frac{5 a^{3}-10 a}{25 a^{3}}\)

Simplify each rational expression. $$ \frac{b^{2}-a^{2}}{a-b} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{3}{m}=2-\frac{m}{m-2}\)

Simplify each complex fraction. $$ \frac{\frac{3}{s^{2}}+\frac{7}{s t}+\frac{2}{t^{2}}}{\frac{2}{t^{2}}-\frac{5}{s t}-\frac{3}{s^{2}}} $$

Add or subtract, and then simplify, if possible. See Example 7. $$\frac{3 s}{s-x}+\frac{1}{x-s}$$

Solve each proportion. $$ \frac{a}{a-3}=\frac{5}{a-3} $$

Perform each division. \(\frac{24 b^{7}-32 b^{2}}{16 b^{5}}\)

Simplify each rational expression. $$ \frac{d^{2}-16 c^{2}}{4 c-d} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{x+2}{2 x-6}+\frac{3}{3-x}=\frac{x}{2}\)

Simplify each complex fraction. $$ \frac{5 x y}{1+\frac{1}{x y}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{5 x}{x+1}+\frac{3}{x+1}-\frac{2 x}{x+1}$$

Solve each proportion. $$ \frac{t^{2}-1}{5}=\frac{1-t^{2}}{2 t} $$

Perform each division. Divide \(2 s^{2}+13 s+5\) by \(2 s+3\)

Divide, and then simplify, if possible. See Example 8. $$ \frac{y^{3}-9 y}{y+2} \div(y-3) $$

Simplify each rational expression. $$ \frac{4-x^{2}}{x^{2}-x-2} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{3}{4 x-8}=\frac{1}{36}-\frac{2}{6-3 x}\)

Simplify each complex fraction. $$ \frac{3 a}{a+\frac{1}{a}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{4}{a+4}-\frac{2 a}{a+4}+\frac{3 a}{a+4}$$

Solve each proportion. $$ \frac{n^{2}}{6}=\frac{n}{n-1} $$

Perform each division. Divide \(4 s^{2}+6 s+1\) by \(2 s-1\)

Divide, and then simplify, if possible. See Example 8. $$ \frac{x-2}{x} \div\left(x^{2}-4\right) $$

Simplify each rational expression. $$ \frac{x^{2}-2 x-15}{25-x^{2}} $$

Use the factor theorem and determine whether the first expression is a factor of \(P(x) .\) See Example 5. $$ x-3 ; P(x)=x^{3}-3 x^{2}+5 x-15 $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{2}{x}+\frac{1}{2}=\frac{9}{4 x}-\frac{1}{2 x}\)

Simplify each complex fraction. $$ \frac{\frac{2}{y-1}-\frac{2}{y}}{\frac{3}{y-1}-\frac{1}{1-y}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{1}{x+y}-\frac{1}{x-y}+\frac{2 y}{x^{2}-y^{2}}$$

Solve each proportion. $$ \frac{2.5 x+1}{2}=\frac{4.5}{12} $$

Perform each division. \(\frac{40 m^{17} n^{20}}{35 m^{15} n^{30}}\)

Simplify each rational expression. $$ \frac{20 x^{3}-20 x^{4}}{x^{2}-2 x+1} $$

Use the factor theorem and determine whether the first expression is a factor of \(P(x) .\) See Example 5. \(x+1 ; P(x)=x^{3}+2 x^{2}-2 x-3\) (Hint: Write \(x+1\) as \(x-(-1)\).)

Solve equation. If a solution is extraneous, so indicate. \(\frac{7}{5 x}-\frac{1}{2}=\frac{5}{6 x}+\frac{1}{3}\)

Simplify each complex fraction. $$ \frac{\frac{1}{x}-\frac{4}{x-1}}{\frac{3}{x-1}+\frac{2}{x}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{a}{a-b}+\frac{b}{a+b}-\frac{a^{2}+b^{2}}{a^{2}-b^{2}}$$

Solve each proportion. $$ \frac{2}{5}=\frac{1.5 x-2}{0.25} $$

Perform each division. \(\frac{34 s^{30} t^{15}}{14 s^{40} t^{12}}\)

Simplify each rational expression. $$ \frac{16 m^{5}-2 m^{6}}{m^{2}-16 m+64} $$

Use the factor theorem and determine whether the first expression is a factor of \(P(x) .\) See Example 5. \(x+2 ; P(x)=3 x^{2}-7 x+4\) (Hint: Write as \(x-(-2)\).)

Solve equation. If a solution is extraneous, so indicate. \(\frac{3-5 y}{2+y}=\frac{-5 y-3}{y-2}\)

Simplify each complex fraction. $$ \frac{\frac{t}{x^{2}-y^{2}}}{\frac{t}{x+y}} $$

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{3 x}{2 x-1}+\frac{x+1}{3 x+2}-\frac{2}{6 x^{2}+x-2}$$

Solve each proportion. $$ \frac{t}{10}=\frac{10}{t} $$