# 1.6 Rational expressions  (Page 4/6)

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For the following exercises, divide the rational expressions.

$\frac{3{y}^{2}-7y-6}{2{y}^{2}-3y-9}÷\frac{{y}^{2}+y-2}{2{y}^{2}+y-3}$

$\frac{6{p}^{2}+p-12}{8{p}^{2}+18p+9}÷\frac{6{p}^{2}-11p+4}{2{p}^{2}+11p-6}$

$\frac{p+6}{4p+3}$

$\frac{{q}^{2}-9}{{q}^{2}+6q+9}÷\frac{{q}^{2}-2q-3}{{q}^{2}+2q-3}$

$\frac{18{d}^{2}+77d-18}{27{d}^{2}-15d+2}÷\frac{3{d}^{2}+29d-44}{9{d}^{2}-15d+4}$

$\frac{2d+9}{d+11}$

$\frac{16{x}^{2}+18x-55}{32{x}^{2}-36x-11}÷\frac{2{x}^{2}+17x+30}{4{x}^{2}+25x+6}$

$\frac{144{b}^{2}-25}{72{b}^{2}-6b-10}÷\frac{18{b}^{2}-21b+5}{36{b}^{2}-18b-10}$

$\frac{12b+5}{3b-1}$

$\frac{16{a}^{2}-24a+9}{4{a}^{2}+17a-15}÷\frac{16{a}^{2}-9}{4{a}^{2}+11a+6}$

$\frac{22{y}^{2}+59y+10}{12{y}^{2}+28y-5}÷\frac{11{y}^{2}+46y+8}{24{y}^{2}-10y+1}$

$\frac{4y-1}{y+4}$

$\frac{9{x}^{2}+3x-20}{3{x}^{2}-7x+4}÷\frac{6{x}^{2}+4x-10}{{x}^{2}-2x+1}$

For the following exercises, add and subtract the rational expressions, and then simplify.

$\frac{4}{x}+\frac{10}{y}$

$\frac{10x+4y}{xy}$

$\frac{12}{2q}-\frac{6}{3p}$

$\frac{4}{a+1}+\frac{5}{a-3}$

$\frac{9a-7}{{a}^{2}-2a-3}$

$\frac{c+2}{3}-\frac{c-4}{4}$

$\frac{y+3}{y-2}+\frac{y-3}{y+1}$

$\frac{2{y}^{2}-y+9}{{y}^{2}-y-2}$

$\frac{x-1}{x+1}-\frac{2x+3}{2x+1}$

$\frac{3z}{z+1}+\frac{2z+5}{z-2}$

$\frac{5{z}^{2}+z+5}{{z}^{2}-z-2}$

$\frac{4p}{p+1}-\frac{p+1}{4p}$

$\frac{x}{x+1}+\frac{y}{y+1}$

$\frac{x+2xy+y}{x+xy+y+1}$

For the following exercises, simplify the rational expression.

$\frac{\frac{6}{y}-\frac{4}{x}}{y}$

$\frac{\frac{2}{a}+\frac{7}{b}}{b}$

$\frac{2b+7a}{a{b}^{2}}$

$\frac{\frac{x}{4}-\frac{p}{8}}{p}$

$\frac{\frac{3}{a}+\frac{b}{6}}{\frac{2b}{3a}}$

$\frac{18+ab}{4b}$

$\frac{\frac{3}{x+1}+\frac{2}{x-1}}{\frac{x-1}{x+1}}$

$\frac{\frac{a}{b}-\frac{b}{a}}{\frac{a+b}{ab}}$

$a-b$

$\frac{\frac{2x}{3}+\frac{4x}{7}}{\frac{x}{2}}$

$\frac{\frac{2c}{c+2}+\frac{c-1}{c+1}}{\frac{2c+1}{c+1}}$

$\frac{3{c}^{2}+3c-2}{2{c}^{2}+5c+2}$

$\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x}{y}+\frac{y}{x}}$

## Real-world applications

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: $\text{\hspace{0.17em}}\frac{15{x}^{2}-8x-7}{{x}^{2}-2x+1}.$ $\frac{15x+7}{x-1}$

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.

$\frac{x+9}{x-9}$

## Extensions

For the following exercises, perform the given operations and simplify.

$\frac{{x}^{2}+x-6}{{x}^{2}-2x-3}\cdot \frac{2{x}^{2}-3x-9}{{x}^{2}-x-2}÷\frac{10{x}^{2}+27x+18}{{x}^{2}+2x+1}$

$\frac{\frac{3{y}^{2}-10y+3}{3{y}^{2}+5y-2}\cdot \frac{2{y}^{2}-3y-20}{2{y}^{2}-y-15}}{y-4}$

$\frac{1}{y+2}$

$\frac{\frac{4a+1}{2a-3}+\frac{2a-3}{2a+3}}{\frac{4{a}^{2}+9}{a}}$

$\frac{{x}^{2}+7x+12}{{x}^{2}+x-6}÷\frac{3{x}^{2}+19x+28}{8{x}^{2}-4x-24}÷\frac{2{x}^{2}+x-3}{3{x}^{2}+4x-7}$

$4$

## Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

${\left(5-3\cdot 2\right)}^{2}-6$

$-5$

$64÷\left(2\cdot 8\right)+14÷7$

$2\cdot {5}^{2}+6÷2$

53

For the following exercises, solve the equation.

$5x+9=-11$

$2y+{4}^{2}=64$

$y=24$

For the following exercises, simplify the expression.

$9\left(y+2\right)÷3\cdot 2+1$

$3m\left(4+7\right)-m$

$32m$

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

11

0

whole

$\frac{5}{6}$

$\sqrt{11}$

irrational

## Exponents and Scientific Notation

For the following exercises, simplify the expression.

${2}^{2}\cdot {2}^{4}$

$\frac{{4}^{5}}{{4}^{3}}$

$16$

${\left(\frac{{a}^{2}}{{b}^{3}}\right)}^{4}$

$\frac{6{a}^{2}\cdot {a}^{0}}{2{a}^{-4}}$

${a}^{6}$

$\frac{{\left(xy\right)}^{4}}{{y}^{3}}\cdot \frac{2}{{x}^{5}}$

$\frac{{4}^{-2}{x}^{3}{y}^{-3}}{2{x}^{0}}$

$\frac{{x}^{3}}{32{y}^{3}}$

${\left(\frac{2{x}^{2}}{y}\right)}^{-2}$

$\left(\frac{16{a}^{3}}{{b}^{2}}\right){\left(4a{b}^{-1}\right)}^{-2}$

$a$

Write the number in standard notation: $\text{\hspace{0.17em}}2.1314\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-6}$

Write the number in scientific notation: 16,340,000

$1.634\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{7}$

For the following exercises, find the principal square root.

$\sqrt{121}$

$\sqrt{196}$

14

$\sqrt{361}$

$\sqrt{75}$

$5\sqrt{3}$

$\sqrt{162}$

$\sqrt{\frac{32}{25}}$

$\frac{4\sqrt{2}}{5}$

$\sqrt{\frac{80}{81}}$

$\sqrt{\frac{49}{1250}}$

$\frac{7\sqrt{2}}{50}$

$\frac{2}{4+\sqrt{2}}$

$4\sqrt{3}+6\sqrt{3}$

$10\sqrt{3}$

$12\sqrt{5}-13\sqrt{5}$

$\sqrt{-243}$

$-3$

$\frac{\sqrt{250}}{\sqrt{-8}}$

## Polynomials

For the following exercises, perform the given operations and simplify.

$\left(3{x}^{3}+2x-1\right)+\left(4{x}^{2}-2x+7\right)$

$3{x}^{3}+4{x}^{2}+6$

$\left(2y+1\right)-\left(2{y}^{2}-2y-5\right)$

$\left(2{x}^{2}+3x-6\right)+\left(3{x}^{2}-4x+9\right)$

$5{x}^{2}-x+3$

$\left(6{a}^{2}+3a+10\right)-\left(6{a}^{2}-3a+5\right)$

$\left(k+3\right)\left(k-6\right)$

${k}^{2}-3k-18$

$\left(2h+1\right)\left(3h-2\right)$

$\left(x+1\right)\left({x}^{2}+1\right)$

${x}^{3}+{x}^{2}+x+1$

$\left(m-2\right)\left({m}^{2}+2m-3\right)$

$\left(a+2b\right)\left(3a-b\right)$

$3{a}^{2}+5ab-2{b}^{2}$

$\left(x+y\right)\left(x-y\right)$

## Factoring Polynomials

For the following exercises, find the greatest common factor.

$81p+9pq-27{p}^{2}{q}^{2}$

$9p$

$12{x}^{2}y+4x{y}^{2}-18xy$

$88{a}^{3}b+4{a}^{2}b-144{a}^{2}$

$4{a}^{2}$

For the following exercises, factor the polynomial.

$2{x}^{2}-9x-18$

$8{a}^{2}+30a-27$

$\left(4a-3\right)\left(2a+9\right)$

${d}^{2}-5d-66$

${x}^{2}+10x+25$

${\left(x+5\right)}^{2}$

${y}^{2}-6y+9$

$4{h}^{2}-12hk+9{k}^{2}$

${\left(2h-3k\right)}^{2}$

$361{x}^{2}-121$

${p}^{3}+216$

$\left(p+6\right)\left({p}^{2}-6p+36\right)$

$8{x}^{3}-125$

$64{q}^{3}-27{p}^{3}$

$\left(4q-3p\right)\left(16{q}^{2}+12pq+9{p}^{2}\right)$

$4x{\left(x-1\right)}^{-\frac{1}{4}}+3{\left(x-1\right)}^{\frac{3}{4}}$

$3p{\left(p+3\right)}^{\frac{1}{3}}-8{\left(p+3\right)}^{\frac{4}{3}}$

${\left(p+3\right)}^{\frac{1}{3}}\left(-5p-24\right)$

$4r{\left(2r-1\right)}^{-\frac{2}{3}}-5{\left(2r-1\right)}^{\frac{1}{3}}$

## Rational Expressions

For the following exercises, simplify the expression.

$\frac{{x}^{2}-x-12}{{x}^{2}-8x+16}$

$\frac{x+3}{x-4}$

$\frac{4{y}^{2}-25}{4{y}^{2}-20y+25}$

$\frac{2{a}^{2}-a-3}{2{a}^{2}-6a-8}\cdot \frac{5{a}^{2}-19a-4}{10{a}^{2}-13a-3}$

$\frac{1}{2}$

$\frac{d-4}{{d}^{2}-9}\cdot \frac{d-3}{{d}^{2}-16}$

$\frac{{m}^{2}+5m+6}{2{m}^{2}-5m-3}÷\frac{2{m}^{2}+3m-9}{4{m}^{2}-4m-3}$

$\frac{m+2}{m-3}$

$\frac{4{d}^{2}-7d-2}{6{d}^{2}-17d+10}÷\frac{8{d}^{2}+6d+1}{6{d}^{2}+7d-10}$

$\frac{10}{x}+\frac{6}{y}$

$\frac{6x+10y}{xy}$

$\frac{12}{{a}^{2}+2a+1}-\frac{3}{{a}^{2}-1}$

$\frac{\frac{1}{d}+\frac{2}{c}}{\frac{6c+12d}{dc}}$

$\frac{1}{6}$

$\frac{\frac{3}{x}-\frac{7}{y}}{\frac{2}{x}}$

## Chapter practice test

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

$-13$

rational

$\sqrt{2}$

For the following exercises, evaluate the equations.

$2\left(x+3\right)-12=18$

$x=12$

$y{\left(3+3\right)}^{2}-26=10$

Write the number in standard notation: $3.1415\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{6}$

3,141,500

Write the number in scientific notation: 0.0000000212.

For the following exercises, simplify the expression.

$-2\cdot {\left(2+3\cdot 2\right)}^{2}+144$

$16$

$4\left(x+3\right)-\left(6x+2\right)$

${3}^{5}\cdot {3}^{-3}$

9

${\left(\frac{2}{3}\right)}^{3}$

$\frac{8{x}^{3}}{{\left(2x\right)}^{2}}$

$2x$

$\left(16{y}^{0}\right)2{y}^{-2}$

$\sqrt{441}$

21

$\sqrt{490}$

$\sqrt{\frac{9x}{16}}$

$\frac{3\sqrt{x}}{4}$

$\frac{\sqrt{121{b}^{2}}}{1+\sqrt{b}}$

$6\sqrt{24}+7\sqrt{54}-12\sqrt{6}$

$21\sqrt{6}$

$\frac{\sqrt{-8}}{\sqrt{625}}$

$\left(13{q}^{3}+2{q}^{2}-3\right)-\left(6{q}^{2}+5q-3\right)$

$13{q}^{3}-4{q}^{2}-5q$

$\left(6{p}^{2}+2p+1\right)+\left(9{p}^{2}-1\right)$

$\left(n-2\right)\left({n}^{2}-4n+4\right)$

${n}^{3}-6{n}^{2}+12n-8$

$\left(a-2b\right)\left(2a+b\right)$

For the following exercises, factor the polynomial.

$16{x}^{2}-81$

$\left(4x+9\right)\left(4x-9\right)$

${y}^{2}+12y+36$

$27{c}^{3}-1331$

$\left(3c-11\right)\left(9{c}^{2}+33c+121\right)$

$3x{\left(x-6\right)}^{-\frac{1}{4}}+2{\left(x-6\right)}^{\frac{3}{4}}$

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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