Chapter 6

Simplify each expression. Write answers using positive exponents. $$ \frac{y^{-3} y^{-4} y^{0}}{\left(2 y^{-2}\right)^{3}} $$

Perform each division. Divide \(x^{4}+2+4 x^{2}+3 x+2 x^{3}\) by \(x^{2}+2+x\)

Divide, and then simplify, if possible. See Example 6. $$ \frac{24 a^{6}}{b} \div \frac{64 a^{9}}{b^{2}} $$

Simplify each function. List any restrictions on the domain. $$ f(x)=\frac{x^{2}+6 x-16}{x^{2}-4} $$

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(3) $$

Simplify each complex fraction. $$ \frac{5 a b^{2}}{\frac{a b}{25}} $$

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

Solve each proportion. $$ \frac{b+4}{5}=\frac{3 b-6}{3} $$

Simplify each expression. Write answers using positive exponents. $$ \left(-x^{2}\right)^{5} y^{7} y^{3} x^{-2} y^{0} $$

Divide, and then simplify, if possible. See Example 6. $$ \frac{p q^{29}}{50} \div \frac{p^{10} q^{38}}{15} $$

Simplify each function. List any restrictions on the domain. $$ f(x)=\frac{9 x^{2}+81 x}{x^{3}+9 x^{2}} $$

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(0) $$

Simplify each complex fraction. $$ \frac{\frac{6 a^{2} b}{4 t}}{3 a^{2} b^{2}} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{m+6}{3 m-12}+\frac{5}{4-m}=\frac{2}{3}\)

Solve each proportion. $$ \frac{2 y+6}{3}=\frac{4 y-16}{5} $$

Simplify each expression. Write answers using positive exponents. $$ 5^{2} r^{-5}\left(r^{6}\right)^{3} $$

Divide, and then simplify, if possible. See Example 6. $$ \frac{s^{41} t^{3}}{12} \div \frac{s^{41} t^{52}}{144} $$

Simplify each function. List any restrictions on the domain. $$ f(x)=\frac{5 x^{2}+50 x}{x^{5}+10 x^{4}} $$

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(-3) $$

Simplify each complex fraction. $$ \frac{a-4+\frac{1}{a}}{-\frac{1}{a}-a+4} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{3}{y}+\frac{7}{2 y}=13\)

Solve each proportion. $$ \frac{5}{b+3}=\frac{b}{2} $$

Fireplaces. A mason and his assistant work together for 6 hours on a brick fireplace before the mason has to leave the job. The assistant finishes the job alone in 10 hours. If the mason can construct a fireplace in 18 hours working alone, how long does it take his assistant working alone to construct a fireplace?

Simplify each function. List any restrictions on the domain. $$ g(x)=\frac{x^{3}+27}{x^{3}+3 x^{2}+4 x+12} $$

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

Simplify each complex fraction. $$ \frac{a+1+\frac{1}{a^{2}}}{\frac{1}{a^{2}}+a-1} $$

Solve each proportion. $$ \frac{p+2}{p+5}=\frac{p-3}{p-2} $$

Extended Vacation. Use the facts in the text message to determine how long the student had originally planned to stay in Europe. Hi Mom, After working so hard to earn \(\$ 1,200\) to take this trip to Europe, it was all worth it! I've been very thrifty and been able to cut \(\$ 20\) from my daily expenses. Because of this, I'll be able to stay three extra days. Please pick me up on Friday instead. Love, Liz

Divide, and then simplify, if possible. See Example 7. $$ \frac{x^{9}}{x^{3}+125} \div \frac{x^{4}}{x^{2}+5 x} $$

Simplify each function. List any restrictions on the domain. $$ g(x)=\frac{x^{3}+64}{x^{3}+4 x^{2}+3 x+12} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{3}{r}+\frac{12}{r^{2}-4 r}=-\frac{7}{r-4}\)

Simplify each complex fraction. $$ \frac{\frac{2}{a+1}+\frac{1}{a+3}}{\frac{2 a}{a^{2}+4 a+3}} $$

Solve each proportion. $$ \frac{9 z+6}{z^{2}+3 z}=\frac{7}{z+3} $$

Perform each division. \(\frac{4 a^{3}+a^{2}-3 a+7}{a+1}\)

Divide, and then simplify, if possible. See Example 7. $$ \frac{a^{2}-18 a+81}{a^{40}} \div \frac{(a-9)^{3}}{a^{37}} $$

Simplify each function. List any restrictions on the domain. $$ s(a)=\frac{a^{3}-a^{2}-6 a+6}{a^{3}-1} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{4 t^{2}+36}{t^{2}-9}-\frac{4 t}{t+3}=\frac{-12}{t-3}\)

Simplify each complex fraction. $$ \frac{\frac{2}{t-2}-\frac{1}{t^{2}+t-6}}{\frac{4}{t+3}} $$

Perform the operations and simplify the result when possible. See Example 6 $$\frac{x-1}{4 x-24}-\frac{3 x-2}{5 x-30}$$

Solve each proportion. $$ \frac{3}{n^{2}+3 n}=\frac{2}{n^{2}+4 n+3} $$

Perform each division. \(\frac{3 x^{3}-2 x^{2}+x-6}{x-1}\)

Simplify each function. List any restrictions on the domain. $$ h(t)=\frac{t^{3}-5 t^{2}-5 t+25}{t^{3}-125} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{x+4}{2 x+14}-\frac{x}{2 x+6}=\frac{3}{16}\)

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

Add or subtract, and then simplify, if possible. See Example 7. $$\frac{5 x}{x-3}+\frac{4 x}{3-x}$$

Solve each proportion. $$ \frac{h^{2}}{5}=\frac{h}{2 h-9} $$

Perform each division. \(\left(x^{6}-x^{4}+2 x^{2}-8\right) \div\left(x^{2}-2\right)\)

Divide, and then simplify, if possible. See Example 7. $$ \frac{3 n^{2}+5 n-2}{12 n^{2}-13 n+3} \div \frac{n^{2}+3 n+2}{4 n^{2}+5 n-6} $$

Simplify each rational expression. $$ \frac{3 m^{2}-2 m n-n^{2}}{m n-m^{2}} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{30}{y-2}+\frac{24}{y-5}=13\)