Deck 6: Rational Expressions and Equations

Question 1

Find all numbers for which the rational expression is not defined.

- $\frac{2}{a-3}$

A) None
B) $\mathrm{a}=3$
C) $\mathrm{a}=0$
D) $\mathrm{a}=-3$

Answer 1

$\mathrm{a}=3$

Question 2

Find all numbers for which the rational expression is not defined.

- $\frac{6}{b+7}$

A) $b=-7$
B) $b=0$
C) None
D) $b=7$

Answer 2

$b=-7$

Question 3

Find all numbers for which the rational expression is not defined.

- $\frac{\mathrm{C}-6}{3}$

A) None
B) $\mathrm{c}=6$
C) $\mathrm{c}=0$
D) $\mathrm{c}=-6$

Answer 3

None

Question 4

Find all numbers for which the rational expression is not defined.

- $\frac{d-5}{8-d}$

A) $d=8,5$
B) None
C) $d=-8$
D) $d=8$

Question 5

Find all numbers for which the rational expression is not defined.

- $\frac{x^{2}-16}{x^{2}-7 x+10}$

A) $x=-5$ and $x=-2$
B) $x=0$
C) $x=4$ and $x=-4$
D) $x=5$ and $x=2$

Question 6

Find all numbers for which the rational expression is not defined.

- $\frac{x^{2}-4}{x^{2}-5 x-24}$

A) $x=3$ and $x=-8$
B) $x=0$
C) $x=2$ and $x=-2$
D) $x=-3$ and $x=8$

Question 7

Find all numbers for which the rational expression is not defined.

- $\frac{x^{2}-16}{x^{2}-5 x-24}$

A) $x=-8$ and $x=3$
B) $x=8$ and $x=-3$
C) $x=0$
D) $x=4$ and $x=-4$

Question 8

Find all numbers for which the rational expression is not defined.

- $\frac{5 y-4}{y^{2}-49}$

A) $y=7$ and $y=-7$
B) $y=\frac{4}{5}$
C) $y=7$
D) $y=49$

Question 9

Find all numbers for which the rational expression is not defined.

- $\frac{b-7}{7 b-21}$

A) $b=21$
B) $b=-7$
C) $b=0$
D) $b=3$

Question 10

Multiply. Do not simplify.

- $\frac{7 y^{4}}{7 y^{4}} \cdot \frac{4 z}{7 x}$

A) $\frac{\left(7 y^{4}\right)(4 z)}{7 x}$
B) $\frac{4 z}{\left(7 y^{4}\right)(7 x)}$
C) $\frac{\left(7 y^{4}\right)(4 z)}{\left(7 y^{4}\right)(7 x)}$
D) $\frac{\left(7 y^{4}\right)(7 x)}{\left(7 y^{4}\right)(4 z)}$

Question 11

Multiply. Do not simplify.

- $\frac{3 \mathrm{x}}{3 \mathrm{x}} \cdot \frac{\mathrm{x}-5}{\mathrm{x}+6}$

A) $\frac{3 x(x-5)}{3 x(x+6)}$
B) $\frac{3 x(x+6)}{3 x(x-5)}$
C) $\frac{3 x(x-5)}{x+6}$
D) $\frac{x-5}{3 x(x+6)}$

Question 12

Multiply. Do not simplify.

- $\frac{5 a+1}{8 a-1} \cdot \frac{a}{a}$

A) $\frac{5 a+1}{(8 a-1) a}$
B) $\frac{(5 a+1) a}{(8 a-1) a}$
C) $\frac{(8 a-1) a}{(5 a+1) a}$
D) $\frac{(5 a+1) a}{8 a-1}$

Question 13

Multiply. Do not simplify.

- $\frac{x-6}{4-x} \cdot \frac{-1}{-1}$

A) $\frac{(x-6)(-1)}{(4-x)(-1)}$
B) $\frac{(x-6)(-1)}{4-x}$
C) $\frac{x-6}{(4-x)(-1)}$
D) $\frac{(4-x)(-1)}{(x-6)(-1)}$

Question 14

Multiply. Do not simplify.

- $\frac{x+2}{x+6} \cdot \frac{x-2}{x-2}$

A) $\frac{x+2}{(x+6)(x-2)}$
B) $\frac{(\mathrm{x}+2)(\mathrm{x}-2)}{\mathrm{x}+6}$
C) $\frac{(\mathrm{x}+2)(\mathrm{x}-2)}{(\mathrm{x}+6)(\mathrm{x}-2)}$
D) $\frac{(x+6)(x-2)}{(x+2)(x-2)}$

Question 15

Simplify.

- $\frac{15 \mathrm{k}^{3}}{5 \mathrm{k}}$

A) $10 \mathrm{k}^{2}$
B) $3 k^{2}$
C) $3 \mathrm{k}$
D) 10

Question 16

Simplify.

- $\frac{32 m^{4} p^{2}}{8 m^{8} p}$

A) $\frac{4 p}{m^{4}}$
B) $\frac{4 m^{4}}{p}$
C) $4 m^{4} p^{2}$
D) $4 \mathrm{mp}$

Question 17

Simplify.

- $\frac{20 x+15}{25}$

A) $\frac{4 x}{5}$
B) $4 x+3$
C) $\frac{4 x+3}{5}$
D) $\frac{4 x+3}{25}$

Question 18

Simplify.

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

A) $\frac{m+1}{3 m+2}$
B) $\frac{m^{2}+1}{3 m^{2}+2}$
C) $\frac{m+1}{m+2}$
D) $\frac{m+3}{3 m+2}$

Question 19

Simplify.

- $\frac{9 x^{2}-36 x}{15 x^{2}-60 x}$

A) 3
B) $\frac{3}{5}$
C) $\frac{5}{3}$
D) $\frac{x-4}{5 x-20}$

Question 20

Simplify.

- $\frac{8-6 x}{3 x-4}$

A) $6 x-8$
B) $3 x-4$
C) -2
D) 2

Question 21

Simplify.

- $\frac{y^{2}+2 y}{y^{2}+8 y}$

A) $\frac{7 y+10}{13 y+40}$
B) $\frac{y+2}{y^{2}+8 y}$
C) $\frac{y+2}{y+8}$
D) $\frac{y^{2}+2 y}{y+8}$

Question 22

Simplify.

- $\frac{3 x+2}{12 x^{2}+23 x+10}$

A) $\frac{3 x+4}{4 x+23}$
B) $\frac{1}{4 x+5}$
C) $\frac{3 x+2}{12 x^{2}+23 x+10}$
D) $\frac{3 \mathrm{x}}{4 \mathrm{x}+5}$

Question 23

Simplify.

- $\frac{y^{2}+2 y-8}{y^{2}+3 y-10}$

A) $\frac{2 y-8}{3 y-10}$
B) $\frac{2 y-4}{3 y-5}$
C) $-\frac{y^{2}+2 y-8}{y^{2}+3 y-10}$
D) $\frac{y+4}{y+5}$

Question 24

Simplify.

- $\frac{y^{2}-2 y-35}{y^{2}+14 y+45}$

A) $-\frac{y^{2}-2 y-35}{y^{2}+14 y+45}$
B) $\frac{-2 y-35}{14 y+45}$
C) $\frac{-2 y+35}{14 y-14}$
D) $\frac{y-7}{y+9}$

Question 25

Simplify.

- $\frac{y^{2}-4 y-12}{y^{2}-5 y-14}$

A) $-\frac{y^{2}-4 y-12}{y^{2}-5 y-14}$
B) $\frac{-4 y-12}{-5 y-14}$
C) $\frac{-4 y-6}{-5 y-7}$
D) $\frac{y-6}{y-7}$

Question 26

Multiply and, if possible, simplify.

- $\frac{2 z^{3}}{4} \cdot \frac{12}{z^{2}}$

A) $6 z$
B) $\frac{6 z^{2}}{z^{3}}$
C) $\frac{6}{\mathrm{z}}$
D) $\frac{z}{6}$

Question 27

Multiply and, if possible, simplify.

- $\frac{8 x^{2}}{7} \cdot \frac{14}{x^{3}}$

A) $\frac{16}{x}$
B) $\frac{x^{5}}{12}$
C) $\frac{x}{12}$
D) $\frac{16 x^{2}}{x^{3}}$

Question 28

Multiply and, if possible, simplify.

- $\frac{2 p-2}{p} \cdot \frac{8 p^{2}}{6 p-6}$

A) $\frac{16 p^{3}-16 p^{2}}{6 p^{2}-6 p}$
B) $\frac{8 p}{3}$
C) $\frac{12 p^{2}+24 p+12}{8 p^{3}}$
D) $\frac{3}{8 p}$

Question 29

Multiply and, if possible, simplify.

- $\frac{\mathrm{k}^{2}+14 \mathrm{k}+45}{\mathrm{k}^{2}+18 \mathrm{k}+81} \cdot \frac{\mathrm{k}^{2}+9 \mathrm{k}}{\mathrm{k}^{2}+11 \mathrm{k}+30}$

A) $\frac{\mathrm{k}}{\mathrm{k}+6}$
B) $\frac{\mathrm{k}^{2}+9 \mathrm{k}}{\mathrm{k}+6}$
C) $\frac{1}{\mathrm{k}+6}$
D) $\frac{\mathrm{k}}{\mathrm{k}^{2}+18 \mathrm{k}+81}$

Question 30

Multiply and, if possible, simplify.

- $\frac{\mathrm{k}^{2}+9 \mathrm{k}+18}{\mathrm{k}^{2}+14 \mathrm{k}+48} \cdot \frac{\mathrm{k}^{2}+8 \mathrm{k}}{\mathrm{k}^{2}-4 \mathrm{k}-21}$

A) $\frac{k^{2}+8 k}{k-7}$
B) $\frac{\mathrm{k}}{\mathrm{k}^{2}+14 \mathrm{k}+48}$
C) $\frac{1}{\mathrm{k}-7}$
D) $\frac{\mathrm{k}}{\mathrm{k}-7}$

Question 31

Multiply and, if possible, simplify.

- $\frac{\mathrm{x}^{2}}{\mathrm{x}^{2}-64} \cdot \frac{\mathrm{x}^{2}-13 \mathrm{x}+40}{\mathrm{x}^{2}-5 \mathrm{x}}$

A) $\frac{x}{x-8}$
B) $\frac{x}{x+8}$
C) $\frac{x^{2}}{x+8}$
D) $\frac{x}{(x+8)(x+5)}$

Question 32

Multiply and, if possible, simplify.

- $\frac{\mathrm{x}^{4}-1}{\mathrm{x}^{4}-81} \cdot \frac{\mathrm{x}^{2}+9}{\mathrm{x}^{2}+1}$

A) $\frac{x+1}{x-3}$
B) $\frac{x-1}{x+3}$
C) $\frac{(\mathrm{x}+1)(\mathrm{x}-1)}{(\mathrm{x}+3)(\mathrm{x}-3)}$
D) $\frac{x-1}{x-3}$

Question 33

Find the reciprocal.

- $\frac{10}{y}$

A) $-\frac{10}{\mathrm{y}}$
B) $\frac{1}{10 y}$
C) $1-\frac{10}{\mathrm{y}}$
D) $\frac{y}{10}$

Question 34

Find the reciprocal.

- $\frac{x}{2}$

A) $-\frac{x}{2}$
B) $\frac{1}{2 \mathrm{x}}$
C) $1-\frac{x}{2}$
D) $\frac{2}{x}$

Question 35

Find the reciprocal.

- $\frac{t+8}{t-2}$

A) $\frac{t-2}{t+8}$
B) $\frac{t-8}{t+2}$
C) $\frac{t+2}{t-8}$
D) $\frac{8+t}{2-t}$

Question 36

Find the reciprocal.

- $6 y^{2}+3 y$

A) $3 y^{2}+6 y$
B) $\frac{6 y^{2}}{3 y}$
C) $1-\left(6 y^{2}+3 y\right)$
D) $\frac{1}{6 y^{2}+3 y}$

Question 37

Find the reciprocal.

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

A) $\frac{1}{2 x^{2}+y^{2}}$
B) $-\frac{2}{x^{2}+y^{2}}$
C) $\frac{x^{2}+y^{2}}{2}$
D) $\frac{1}{2 x^{2}+2 y^{2} x}$

Question 38

Find the reciprocal.

- $\frac{x^{2}+6}{4-x}$

A) $\frac{4-x}{x^{2}+6}$
B) $\frac{6-x}{x^{2}+4}$
C) $\frac{x^{2}+6}{x-4}$
D) $\frac{x^{2}+4}{6-x}$

Question 39

Find the reciprocal.

- $\frac{y^{2}+5 y+8}{y^{2}-y+4}$

A) $\frac{y^{2}+5 y-8}{y^{2}-y-4}$
B) $\frac{y^{2}-y+4}{y^{2}+5 y+8}$
C) $\frac{y^{2}-8 y+5}{y^{2}+y+4}$
D) $\frac{y^{2}-y-4}{y^{2}+5 y-8}$

Question 40

Find the reciprocal.

- $\frac{x^{2}+6 x y-y^{2}}{x^{2}-4 x y+y^{2}}$

A) $\frac{x^{2}-4 x y+y^{2}}{x^{2}+6 x y-y^{2}}$
B) $\frac{x^{2}-6 x y+y^{2}}{x^{2}+4 x y-y^{2}}$
C) $\frac{x^{2}+4 x y+y^{2}}{x^{2}-6 x y-y^{2}}$
D) $\frac{1}{x^{2}-4 x y-y^{2}}$

Question 41

Divide and simplify.

- $\frac{3}{8} \div \frac{6}{5}$

A) $\frac{9}{20}$
B) $\frac{5}{16}$
C) $\frac{20}{9}$
D) $\frac{16}{5}$

Question 42

Divide and simplify.

- $\frac{2 x^{2}}{4} \div \frac{x^{3}}{20}$

A) $\frac{x}{10}$
B) $\frac{10 x^{2}}{x^{3}}$
C) $\frac{10}{\mathrm{x}}$
D) $\frac{x^{5}}{40}$

Question 43

Divide and simplify.

- $\frac{6 p-6}{p} \div \frac{7 p-7}{8 p^{2}}$

A) $\frac{7}{48 p}$
B) $\frac{42 p^{2}+84 p+42}{8 p^{3}}$
C) $\frac{48 p}{7}$
D) $\frac{48 p^{3}-48 p^{2}}{7 p^{2}-7 p}$

Question 44

Divide and simplify.

- $\frac{28}{y-x} \div \frac{7}{x-y}$

A) $\frac{4(x-y)}{y-x}$
B) -28
C) 4
D) -4

Question 45

Divide and simplify.

- $\frac{-9+3 x}{9} \div \frac{21-7 x}{21}$

A) $\frac{x-3}{x+3}$
B) - 1
C) 1
D) $\frac{x-3}{3-x}$

Question 46

Divide and simplify.

- $\frac{x^{2}-16}{x} \div \frac{5 x+20}{x-4}$

A) $\frac{5 x}{(x-4)^{2}}$
B) $\frac{(\mathrm{x}-4)^{2}}{\mathrm{x}}$
C) $\frac{5(x+4)^{2}}{x}$
D) $\frac{(x-4)^{2}}{5 x}$

Question 47

Divide and simplify.

- $\frac{x^{2}+9 x+18}{x+3} \div \frac{x^{2}-36}{3 x-18}$

A) $\frac{(x+6)^{2}}{3}$
B) $\frac{3 x-18}{x-6}$
C) $\frac{1}{3}$
D) 3

Question 48

Divide and simplify.

- $\frac{z^{2}+8 z+15}{z^{2}+9 z+20} \div \frac{z^{2}+3 z}{z^{2}+14 z+40}$

A) $\mathrm{z}+10$
B) $\frac{\mathrm{z}+10}{\mathrm{Z}}$
C) $\frac{z+10}{z^{2}+4 z}$
D) $\frac{z}{z^{2}+9 z+20}$

Question 49

Divide and simplify.

- $\frac{z^{2}+10 z+25}{z^{2}+11 z+30} \div \frac{z^{2}+5 z}{z^{2}-3 z-54}$

A) $\frac{z-9}{z^{2}+6 z}$
B) $\mathrm{z}$ - 9
C) $\frac{\mathrm{z}}{\mathrm{z}^{2}+11 \mathrm{z}+30}$
D) $\frac{\mathrm{z}-9}{\mathrm{z}}$

Question 50

Divide and simplify.

- $\frac{a^{2}-9 a+20}{a^{2}+8 a+16} \div \frac{a^{2}-10 a+25}{a^{2}+2 a-24}$

A) $\frac{(a-4)^{2}(a+6)}{(a+4)^{2}}$
B) $-\frac{a+6}{a-5}$
C) $\frac{(a-5)^{3}}{(a+4)^{2}(a+6)}$
D) $\frac{(a-4)^{2}(a+6)}{(a+4)^{2}(a-5)}$

Question 51

Find the LCM.

-36,27

A) 12
B) 108
C) 54
D) 36

Question 52

Find the LCM.

- $20,12,15$

A) 30
B) 60
C) 12
D) 20

Question 53

Find the LCM.

- $30,12,90$

A) 90
B) 36
C) 60
D) 180

Question 54

Add, first finding the LCD. Simplify, if possible.

- $\frac{6}{7}+\frac{2}{9}$

A) $\frac{17}{4}$
B) $\frac{8}{63}$
C) $\frac{68}{63}$
D) $\frac{1}{2}$

Question 55

Add, first finding the LCD. Simplify, if possible.

- $\frac{10}{11}+\frac{5}{6}$

A) $\frac{115}{17}$
B) $\frac{15}{17}$
C) $\frac{115}{66}$
D) $\frac{5}{22}$

Question 56

Add, first finding the LCD. Simplify, if possible.

- $\frac{1}{24}+\frac{7}{18}$

A) $\frac{1}{54}$
B) $\frac{31}{72}$
C) $\frac{4}{21}$
D) $\frac{31}{7}$

Question 57

Add, first finding the LCD. Simplify, if possible.

- $\frac{1}{4}+\frac{4}{7}+\frac{4}{5}$

A) $\frac{227}{16}$
B) $\frac{227}{140}$
C) $\frac{9}{140}$
D) $\frac{9}{16}$

Question 58

Find the LCM.

- $50 x^{5} y, 30 x^{4} y^{3}, 15 x^{2} y^{4}$

A) $50 x^{5} y^{4}$
B) $30 x^{5} y$
C) $150 x^{5} y^{4}$
D) $75 x^{20} y^{12}$

Question 59

Find the LCM.

- $35 x, 5 x^{2}, 7 x^{3}$

A) $7 x^{3}$
B) $35 x^{3}$
C) $35 x^{5}$
D) $35 x^{2}$

Question 60

Find the LCM.

- $6 x y, 15 x^{2}$

A) $30 x^{3} y$
B) $30 x y^{3}$
C) $30 x^{2} y$
D) $30 x y^{2}$

Question 61

Find the LCM.

- $t, t+8$

A) $\mathrm{t}$
B) $t(t+8)$
C) $t+8$
D) 28

Question 62

Find the LCM.

- $7 a+21, a^{2}+3 a$

A) $7 a(a+3)$
B) $7 \mathrm{a}^{2}+3$
C) $7 a+3$
D) $7 \mathrm{a}^{2}+21$

Question 63

Find the LCM.

- $n, 4+n, 4-n$

A) $n^{2}+16$
B) $n(4+n)(4-n)$
C) $16 n^{2}$
D) $n+4$

Question 64

Find the LCM.

- $r^{2}+6 r+9, r^{2}+3 r$

A) $r(r+1)(r+3)$
B) $r(r+3)$
C) $(r+3)^{2}$
D) $r(r+3)^{2}$

Question 65

Find the LCM.

- $m^{2}+2 m, m^{2}+7 m+10$

A) $m(m-3)^{2}$
B) $(m-3)^{2}$
C) $m(m+2)(m+5)$
D) $m(x-3)(m+5)$

Question 66

Find the LCM.

- $x^{2}+10 x+24, x^{2}+8 x+12$

A) $(x+4)(x+6)$
B) $(x-4)(x-6)(x+2)$
C) $(x+4)(x+6)(x+2)$
D) $(x+6)(x+2)$

Question 67

Find the LCM.

- $x^{2}-6 x+5,4 x-20$

A) $4(x+1)(x+5)$
B) $4(x-1)(x+5)$
C) $4(x+1)(x-5)$
D) $4(\mathrm{x}-1)(\mathrm{x}-5)$

Question 68

Add. Simplify, if possible.

- $\frac{6}{8}+\frac{4}{8}$

A) 5
B) $\frac{5}{2}$
C) $\frac{4}{5}$
D) $\frac{5}{4}$

Question 69

Add. Simplify, if possible.

- $\frac{6}{11 \mathrm{x}}+\frac{2}{11 \mathrm{x}}$

A) $\frac{11 x}{8}$
B) $\frac{8}{22 \mathrm{x}}$
C) 1
D) $\frac{8}{11 \mathrm{x}}$

Question 70

Add. Simplify, if possible.

- $\frac{10}{8 x^{2}}+\frac{2}{8 x^{2}}$

A) $\frac{2}{3 x^{2}}$
B) $\frac{3}{2 x^{2}}$
C) $\frac{3}{4 x^{4}}$
D) 3

Question 71

Add. Simplify, if possible.

- $\frac{5 x+7}{2 x-1}+\frac{6-6 x}{-1+2 x}$

A) $\frac{-1 x+13}{2 x-1}$
B) $\frac{-1 \mathrm{x}+13}{4 \mathrm{x}-2}$
C) $\frac{13 x+1}{2 x-1}$
D) $-\frac{1}{2} \mathrm{x}-13$

Question 72

Add. Simplify, if possible.

- $\frac{3}{x+y}+\frac{5}{y+x}$

A) $\frac{-2}{x+y}$
B) $\frac{8}{x+y}$
C) $\frac{3}{x}+\frac{5}{y}$
D) $\frac{8}{2 x+2 y}$

Question 73

Add. Simplify, if possible.

- $\frac{m^{2}-8 m}{m-2}+\frac{12}{m-2}$

A) $m+6$
B) $\mathrm{m}$ - 2
C) $\frac{m^{2}-8 m+12}{m-2}$
D) $m-6$

Question 74

Add. Simplify, if possible.

- $\frac{5}{x-7}+\frac{7}{7-x}$

A) $\frac{-2}{x-7}$
B) $\frac{35}{x-7}$
C) $\frac{2}{x-7}$
D) $\frac{12}{x-7}$

Question 75

Add. Simplify, if possible.

- $\frac{5}{9-m}+\frac{4}{m-9}$

A) $\frac{20}{9-m}$
B) $\frac{-1}{9-m}$
C) $\frac{9}{9-m}$
D) $\frac{1}{9-m}$

Question 76

Add. Simplify, if possible.

- $\frac{8}{m-n^{2}}+\frac{6}{n^{2}-m}$

A) $\frac{2}{m-n^{2}}$
B) $\frac{14}{m-n^{2}}$
C) $\frac{48}{m-n^{2}}$
D) $\frac{-2}{m-n^{2}}$

Question 77

Add. Simplify, if possible.

- $\frac{4}{3 x-5}+\frac{9}{5-3 x}$

A) $\frac{-5}{3 x-5}$
B) $\frac{13}{3 x-5}$
C) $\frac{-13}{3 x-5}$
D) $\frac{5}{3 x-5}$

Question 78

Add. Simplify, if possible.

- $\frac{x+8}{x-4}+\frac{x-5}{4-x}+\frac{9(x+1)}{x-4}$

A) $\frac{9 x+14}{x-4}$
B) $\frac{9 x+22}{(x-4)^{3}}$
C) $\frac{9 x+22}{x-4}$
D) $\frac{11 \mathrm{x}+12}{\mathrm{x}-4}$

Question 79

Add. Simplify, if possible.

- $\frac{2}{21 \mathrm{x}}+\frac{4}{15 \mathrm{x}^{2}}$

A) $\frac{6}{21 x+15 x^{2}}$
B) $\frac{2(5 x+14)}{105 x^{2}}$
C) $\frac{6}{315 x^{2}}$
D) $\frac{48}{105 x^{2}}$

Question 80

Add. Simplify, if possible.

- $\frac{4}{r}+\frac{7}{r-5}$

A) $\frac{20 r-11}{r(5-r)}$
B) $\frac{11 r-20}{r(r-5)}$
C) $\frac{20 r-11}{r(r-5)}$
D) $\frac{11 r-20}{r(5-r)}$