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For the following exercises, divide the rational expressions.
$\frac{3{y}^{2}-7y-6}{2{y}^{2}-3y-9}\xf7\frac{{y}^{2}+y-2}{2{y}^{2}+y-3}$
$\frac{6{p}^{2}+p-12}{8{p}^{2}+18p+9}\xf7\frac{6{p}^{2}-11p+4}{2{p}^{2}+11p-6}$
$\frac{p+6}{4p+3}$
$\frac{{q}^{2}-9}{{q}^{2}+6q+9}\xf7\frac{{q}^{2}-2q-3}{{q}^{2}+2q-3}$
$\frac{18{d}^{2}+77d-18}{27{d}^{2}-15d+2}\xf7\frac{3{d}^{2}+29d-44}{9{d}^{2}-15d+4}$
$\frac{2d+9}{d+11}$
$\frac{16{x}^{2}+18x-55}{32{x}^{2}-36x-11}\xf7\frac{2{x}^{2}+17x+30}{4{x}^{2}+25x+6}$
$\frac{144{b}^{2}-25}{72{b}^{2}-6b-10}\xf7\frac{18{b}^{2}-21b+5}{36{b}^{2}-18b-10}$
$\frac{12b+5}{3b\mathrm{-1}}$
$\frac{16{a}^{2}-24a+9}{4{a}^{2}+17a-15}\xf7\frac{16{a}^{2}-9}{4{a}^{2}+11a+6}$
$\frac{22{y}^{2}+59y+10}{12{y}^{2}+28y-5}\xf7\frac{11{y}^{2}+46y+8}{24{y}^{2}-10y+1}$
$\frac{4y\mathrm{-1}}{y+4}$
$\frac{9{x}^{2}+3x-20}{3{x}^{2}-7x+4}\xf7\frac{6{x}^{2}+4x-10}{{x}^{2}-2x+1}$
For the following exercises, add and subtract the rational expressions, and then simplify.
$\frac{12}{2q}-\frac{6}{3p}$
$\frac{c+2}{3}-\frac{c-4}{4}$
$\frac{y+3}{y-2}+\frac{y-3}{y+1}$
$\frac{2{y}^{2}-y+9}{{y}^{2}-y-2}$
$\frac{x-1}{x+1}-\frac{2x+3}{2x+1}$
$\frac{3z}{z+1}+\frac{2z+5}{z-2}$
$\frac{5{z}^{2}+z+5}{{z}^{2}-z-2}$
$\frac{4p}{p+1}-\frac{p+1}{4p}$
For the following exercises, simplify the rational expression.
$\frac{\frac{6}{y}-\frac{4}{x}}{y}$
$\frac{\frac{2}{a}+\frac{7}{b}}{b}$
$\frac{2b+7a}{a{b}^{2}}$
$\frac{\frac{x}{4}-\frac{p}{8}}{p}$
$\frac{\frac{3}{a}+\frac{b}{6}}{\frac{2b}{3a}}$
$\frac{18+ab}{4b}$
$\frac{\frac{3}{x+1}+\frac{2}{x-1}}{\frac{x-1}{x+1}}$
$\frac{\frac{2x}{3}+\frac{4x}{7}}{\frac{x}{2}}$
$\frac{\frac{2c}{c+2}+\frac{c-1}{c+1}}{\frac{2c+1}{c+1}}$
$\frac{3{c}^{2}+3c-2}{2{c}^{2}+5c+2}$
$\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y}+\frac{y}{x}}$
Brenda is placing tile on her bathroom floor. The area of the floor is $\text{\hspace{0.17em}}15{x}^{2}-8x-7\text{\hspace{0.17em}}$ ft ^{2} . The area of one tile is $\text{\hspace{0.17em}}{x}^{2}-2x+1{\text{ft}}^{2}.\text{\hspace{0.17em}}$ To find the number of tiles needed, simplify the rational expression: $\text{\hspace{0.17em}}\frac{15{x}^{2}-8x-7}{{x}^{2}-2x+1}.$
$\frac{15x+7}{x\mathrm{-1}}$
The area of Sandy’s yard is $\text{\hspace{0.17em}}25{x}^{2}-625\text{\hspace{0.17em}}$ ft ^{2} . A patch of sod has an area of $\text{\hspace{0.17em}}{x}^{2}-10x+25\text{\hspace{0.17em}}$ ft ^{2} . Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard.
Aaron wants to mulch his garden. His garden is $\text{\hspace{0.17em}}{x}^{2}+18x+81\text{\hspace{0.17em}}$ ft ^{2} . One bag of mulch covers $\text{\hspace{0.17em}}{x}^{2}-81\text{\hspace{0.17em}}$ ft ^{2} . Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.
$\frac{x+9}{x\mathrm{-9}}$
For the following exercises, perform the given operations and simplify.
$\frac{{x}^{2}+x-6}{{x}^{2}-2x-3}\cdot \frac{2{x}^{2}-3x-9}{{x}^{2}-x-2}\xf7\frac{10{x}^{2}+27x+18}{{x}^{2}+2x+1}$
$\frac{\frac{3{y}^{2}-10y+3}{3{y}^{2}+5y-2}\cdot \frac{2{y}^{2}-3y-20}{2{y}^{2}-y-15}}{y-4}$
$\frac{1}{y+2}$
$\frac{\frac{4a+1}{2a-3}+\frac{2a-3}{2a+3}}{\frac{4{a}^{2}+9}{a}}$
$\frac{{x}^{2}+7x+12}{{x}^{2}+x-6}\xf7\frac{3{x}^{2}+19x+28}{8{x}^{2}-4x-24}\xf7\frac{2{x}^{2}+x-3}{3{x}^{2}+4x-7}$
$4$
For the following exercises, perform the given operations.
$64\xf7\left(2\cdot 8\right)+14\xf77$
For the following exercises, solve the equation.
$5x+9=\mathrm{-11}$
For the following exercises, simplify the expression.
$9\left(y+2\right)\xf73\cdot 2+1$
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.
$\frac{5}{6}$
For the following exercises, simplify the expression.
${2}^{2}\cdot {2}^{4}$
${\left(\frac{{a}^{2}}{{b}^{3}}\right)}^{4}$
$\frac{6{a}^{2}\cdot {a}^{0}}{2{a}^{\mathrm{-4}}}$
${a}^{6}$
$\frac{{\left(xy\right)}^{4}}{{y}^{3}}\cdot \frac{2}{{x}^{5}}$
$\frac{{4}^{\mathrm{-2}}{x}^{3}{y}^{\mathrm{-3}}}{2{x}^{0}}$
$\frac{{x}^{3}}{32{y}^{3}}$
${\left(\frac{2{x}^{2}}{y}\right)}^{\mathrm{-2}}$
$\left(\frac{16{a}^{3}}{{b}^{2}}\right){\left(4a{b}^{\mathrm{-1}}\right)}^{\mathrm{-2}}$
$a$
Write the number in standard notation: $\text{\hspace{0.17em}}2.1314\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{\mathrm{-6}}$
Write the number in scientific notation: 16,340,000
$1.634\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{7}$
For the following exercises, find the principal square root.
$\sqrt{121}$
$\sqrt{361}$
$\sqrt{162}$
$\sqrt{\frac{80}{81}}$
$\frac{2}{4+\sqrt{2}}$
$12\sqrt{5}-13\sqrt{5}$
$\frac{\sqrt[3]{250}}{\sqrt[3]{\mathrm{-8}}}$
For the following exercises, perform the given operations and simplify.
$\left(2y+1\right)-\left(2{y}^{2}-2y-5\right)$
$\left(6{a}^{2}+3a+10\right)-\left(6{a}^{2}\mathrm{-3}a+5\right)$
$(2h+1)(3h-2)$
$\left(x+1\right)\left({x}^{2}+1\right)$
${x}^{3}+{x}^{2}+x+1$
$(m-2)({m}^{2}+2m-3)$
$\left(a+2b\right)\left(3a-b\right)$
$3{a}^{2}+5ab-2{b}^{2}$
$\left(x+y\right)\left(x-y\right)$
For the following exercises, find the greatest common factor.
$12{x}^{2}y+4x{y}^{2}\mathrm{-18}xy$
For the following exercises, factor the polynomial.
$2{x}^{2}-9x-18$
${d}^{2}-5d-66$
${y}^{2}-6y+9$
$361{x}^{2}-121$
$8{x}^{3}-125$
$4x{(x-1)}^{-\frac{1}{4}}+3{(x-1)}^{\frac{3}{4}}$
$3p{\left(p+3\right)}^{\frac{1}{3}}\mathrm{-8}{\left(p+3\right)}^{\frac{4}{3}}$
${\left(p+3\right)}^{\frac{1}{3}}\left(\mathrm{-5}p-24\right)$
$4r{\left(2r-1\right)}^{-\frac{2}{3}}-5{\left(2r-1\right)}^{\frac{1}{3}}$
For the following exercises, simplify the expression.
$\frac{4{y}^{2}-25}{4{y}^{2}-20y+25}$
$\frac{2{a}^{2}-a-3}{2{a}^{2}-6a-8}\cdot \frac{5{a}^{2}-19a-4}{10{a}^{2}-13a-3}$
$\frac{1}{2}$
$\frac{d-4}{{d}^{2}-9}\cdot \frac{d-3}{{d}^{2}-16}$
$\frac{{m}^{2}+5m+6}{2{m}^{2}-5m-3}\xf7\frac{2{m}^{2}+3m-9}{4{m}^{2}-4m-3}$
$\frac{m+2}{m-3}$
$\frac{4{d}^{2}-7d-2}{6{d}^{2}-17d+10}\xf7\frac{8{d}^{2}+6d+1}{6{d}^{2}+7d-10}$
$\frac{12}{{a}^{2}+2a+1}-\frac{3}{{a}^{2}\mathrm{-1}}$
$\frac{\frac{1}{d}+\frac{2}{c}}{\frac{6c+12d}{dc}}$
$\frac{1}{6}$
$\frac{\frac{3}{x}-\frac{7}{y}}{\frac{2}{x}}$
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.
$\sqrt{2}$
For the following exercises, evaluate the equations.
$y{(3+3)}^{2}-26=10$
Write the number in standard notation: $3.1415\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{6}$
3,141,500
Write the number in scientific notation: 0.0000000212.
For the following exercises, simplify the expression.
$4\left(x+3\right)-\left(6x+2\right)$
${\left(\frac{2}{3}\right)}^{3}$
$\left(16{y}^{0}\right)2{y}^{\mathrm{-2}}$
$\sqrt{490}$
$\frac{\sqrt{121{b}^{2}}}{1+\sqrt{b}}$
$\frac{\sqrt[3]{\mathrm{-8}}}{\sqrt[4]{625}}$
$(13{q}^{3}+2{q}^{2}-3)-(6{q}^{2}+5q-3)$
$13{q}^{3}-4{q}^{2}-5q$
$\left(6{p}^{2}+2p+1\right)+\left(9{p}^{2}\mathrm{-1}\right)$
$(a-2b)(2a+b)$
For the following exercises, factor the polynomial.
${y}^{2}+12y+36$
$3x{(x-6)}^{-\frac{1}{4}}+2{(x-6)}^{\frac{3}{4}}$
For the following exercises, simplify the expression.
$\frac{2{z}^{2}+7z+3}{{z}^{2}-9}\cdot \frac{4{z}^{2}-15z+9}{4{z}^{2}-1}$
$\frac{4z-3}{2z-1}$
$\frac{x}{y}+\frac{2}{x}$
$\frac{\frac{a}{2b}-\frac{2b}{9a}}{\frac{3a-2b}{6a}}$
$\frac{3a+2b}{3b}$
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