Deck 12: Rational Expressions and Equations

Question 1

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

- $\frac{2}{a-7}$

A) $\mathrm{a}=0$
B) $\mathrm{a}=7$
C) None
D) $\mathrm{a}=-7$

Answer 1

$\mathrm{a}=7$

Question 2

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

- $\frac{7}{b+8}$

A) $b=0$
B) $b=-8$
C) $b=8$
D) None

Answer 2

$b=-8$

Question 3

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

- $\frac{c-9}{5}$

A) $c=0$
B) $c=-9$
C) None
D) $\mathrm{c}=9$

Answer 3

None

Question 4

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

- $\frac{d-6}{2-d}$

A) $d=2,6$
B) $d=-2$
C) $d=2$
D) None

Question 5

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

- $\frac{x^{2}-36}{x^{2}-13 x+42}$

A) $x=-7$ and $x=-6$
B) $x=0$
C) $x=6$ and $x=-6$
D) $x=7$ and $x=6$

Question 6

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

- $\frac{x^{2}-81}{x^{2}+5 x-14}$

A) $x=7$ and $x=-2$
B) $x=0$
C) $x=9$ and $x=-9$
D) $x=-7$ and $x=2$

Question 7

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

- $\frac{x^{2}-81}{x^{2}-2 x-15}$

A) $x=9$ and $x=-9$
B) $x=0$
C) $x=5$ and $x=-3$
D) $x=-5$ and $x=3$

Question 8

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

- $\frac{2 y-6}{y^{2}-49}$

A) $y=49$
B) $y=7$
C) $y=7$ and $y=-7$
D) $y=3$

Question 9

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

- $\frac{b-7}{7 b-28}$

A) $b=4$
B) $b=28$
C) $b=0$
D) $b=-7$

Question 10

Multiply. Do not simplify.

- $\frac{7 y^{4}}{7 y^{4}} \cdot \frac{2 z}{7 x}$

A) $\frac{2 z}{\left(7 y^{4}\right)(7 x)}$
B) $\frac{\left(7 y^{4}\right)(7 x)}{\left(7 y^{4}\right)(2 z)}$
C) $\frac{\left(7 y^{4}\right)(2 z)}{7 x}$
D) $\frac{\left(7 y^{4}\right)(2 z)}{\left(7 y^{4}\right)(7 x)}$

Question 11

Multiply. Do not simplify.

- $\frac{3 x}{3 x} \cdot \frac{x-3}{x+10}$

A) $\frac{3 x(x-3)}{x+10}$
B) $\frac{3 x(x+10)}{3 x(x-3)}$
C) $\frac{x-3}{3 x(x+10)}$
D) $\frac{3 x(x-3)}{3 x(x+10)}$

Question 12

Multiply. Do not simplify.

- $\frac{7 a+1}{5 a-1} \cdot \frac{a}{a}$

A) $\frac{(7 a+1) a}{5 a-1}$
B) $\frac{(7 a+1) a}{(5 a-1) a}$
C) $\frac{7 a+1}{(5 a-1) a}$
D) $\frac{(5 a-1) a}{(7 a+1) a}$

Question 13

Multiply. Do not simplify.

- $\frac{x-9}{7-x} \cdot \frac{-1}{-1}$

A) $\frac{(7-x)(-1)}{(x-9)(-1)}$
B) $\frac{(x-9)(-1)}{(7-x)(-1)}$
C) $\frac{x-9}{(7-x)(-1)}$
D) $\frac{(x-9)(-1)}{7-x}$

Question 14

Multiply. Do not simplify.

- $\frac{x+4}{x+9} \cdot \frac{x-4}{x-4}$

A) $\frac{(x+9)(x-4)}{(x+4)(x-4)}$
B) $\frac{(x+4)(x-4)}{(x+9)(x-4)}$
C) $\frac{x+4}{(x+9)(x-4)}$
D) $\frac{(x+4)(x-4)}{x+9}$

Question 15

Simplify.

- $\frac{24 k^{3}}{6 k}$

A) 18
B) $4 \mathrm{k}^{2}$
C) $18 \mathrm{k}^{2}$
D) $4 \mathrm{k}$

Question 16

Simplify.

- $\frac{16 m^{2} p^{2}}{8 m^{9} p}$

A) $\frac{2 m^{7}}{p}$
B) $2 \mathrm{mp}$
C) $\frac{2 p}{m^{7}}$
D) $2 m^{7} p^{2}$

Question 17

Simplify.

- $\frac{16 x-4}{20}$

A) $\frac{4 x-1}{5}$
B) $\frac{4 x}{5}$
C) $\frac{4 x-1}{20}$
D) $4 x-1$

Question 18

Simplify.

- $\frac{3 m^{2}+3 m}{15 m^{2}+21 m}$

A) $\frac{m^{2}+1}{5 m^{2}+7}$
B) $\frac{m+1}{5 m+7}$
C) $\frac{m+1}{m+7}$
D) $\frac{m+3}{5 m+7}$

Question 19

Simplify.

- $\frac{4 x^{2}-20 x}{12 x^{2}-60 x}$

A) 2
B) $\frac{x-5}{6 x-30}$
C) 3
D) $\frac{1}{3}$

Question 20

Simplify.

- $\frac{8-2 x}{x-4}$

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

Question 21

Simplify.

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

A) $\frac{12 y+27}{14 y+45}$
B) $\frac{y^{2}+3 y}{y+5}$
C) $\frac{y+3}{y+5}$
D) $\frac{y+3}{y^{2}+5 y}$

Question 22

Simplify.

- $\frac{4 x+2}{20 x^{2}+26 x+8}$

A) $\frac{4 x+5}{5 x+26}$
B) $\frac{4 x+2}{20 x^{2}+26 x+8}$
C) $\frac{4 x}{5 x+4}$
D) $\frac{1}{5 x+4}$

Question 23

Simplify.

- $\frac{y^{2}-2 y-8}{y^{2}+3 y-28}$

A) $\frac{-2 y-8}{3 y-28}$
B) $\frac{-2 y-2}{3 y-7}$
C) $\frac{y+2}{y+7}$
D) $-\frac{y^{2}-2 y-8}{y^{2}+3 y-28}$

Question 24

Simplify.

- $\frac{y^{2}+3 y-18}{y^{2}+12 y+36}$

A) $\frac{3 y-18}{12 y+36}$
B) $-\frac{y^{2}+3 y-18}{y^{2}+12 y+36}$
C) $\frac{y-3}{y+6}$
D) $\frac{3 y+18}{12 y-12}$

Question 25

Simplify.

- $\frac{y^{2}+3 y-18}{y^{2}+2 y-24}$

A) $-\frac{y^{2}+3 y-18}{y^{2}+2 y-24}$
B) $\frac{3 y-3}{2 y-4}$
C) $\frac{y-3}{y-4}$
D) $\frac{3 y-18}{2 y-24}$

Question 26

Multiply and, if possible, simplify.

- $\frac{2 z^{3}}{3} \cdot \frac{9}{z^{2}}$

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

Question 27

Multiply and, if possible, simplify.

- $\frac{3 x^{2}}{5} \cdot \frac{10}{x^{3}}$

A) $\frac{x}{16}$
B) $\frac{6 x^{2}}{x^{3}}$
C) $\frac{x^{5}}{16}$
D) $\frac{6}{x}$

Question 28

Multiply and, if possible, simplify.

- $\frac{5 p-5}{p} \cdot \frac{5 p^{2}}{6 p-6}$

A) $\frac{6}{25 p}$
B) $\frac{25 p^{3}-25 p^{2}}{6 p^{2}-6 p}$
C) $\frac{25 p}{6}$
D) $\frac{30 p^{2}+60 p+30}{5 p^{3}}$

Question 29

Multiply and, if possible, simplify.

- $\frac{\mathrm{k}^{2}+5 \mathrm{k}+6}{\mathrm{k}^{2}+11 \mathrm{k}+18} \cdot \frac{\mathrm{k}^{2}+9 \mathrm{k}}{\mathrm{k}^{2}+12 \mathrm{k}+27}$

A) $\frac{\mathrm{k}}{\mathrm{k}+9}$
B) $\frac{1}{k+9}$
C) $\frac{k^{2}+9 k}{k+9}$
D) $\frac{\mathrm{k}}{\mathrm{k}^{2}+11 \mathrm{k}+18}$

Question 30

Multiply and, if possible, simplify.

- $\frac{k^{2}+14 k+48}{k^{2}+16 k+64} \cdot \frac{k^{2}+8 k}{k^{2}+3 k-18}$

A) $\frac{k}{k^{2}+16 k+64}$
B) $\frac{k^{2}+8 k}{k-3}$
C) $\frac{\mathrm{k}}{\mathrm{k}-3}$
D) $\frac{1}{k-3}$

Question 31

Multiply and, if possible, simplify.

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

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

Question 32

Find the reciprocal.

- $\frac{10}{y}$

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

Question 33

Find the reciprocal.

- $\frac{x}{7}$

A) $\frac{7}{x}$
B) $\frac{1}{7 x}$
C) $-\frac{x}{7}$
D) $1-\frac{x}{7}$

Question 34

Find the reciprocal.

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

A) $\frac{8+\mathrm{t}}{4-\mathrm{t}}$
B) $\frac{t-8}{t+4}$
C) $\frac{t+4}{t-8}$
D) $\frac{t-4}{t+8}$

Question 35

Find the reciprocal.

- $4 y^{2}+5 y$

A) $1-\left(4 y^{2}+5 y\right)$
B) $\frac{1}{4 y^{2}+5 y}$
C) $5 y^{2}+4 y$
D) $\frac{4 y^{2}}{5 y}$

Question 36

Find the reciprocal.

- $\frac{12}{x^{2}+y^{2}}$

A) $\frac{1}{12 x^{2}+y^{2}}$
B) $-\frac{12}{x^{2}+y^{2}}$
C) $\frac{1}{12 x^{2}+12 y^{2} x}$
D) $\frac{x^{2}+y^{2}}{12}$

Question 37

Find the reciprocal.

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

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

Question 38

Find the reciprocal.

- $\frac{y^{2}+12 y+3}{y^{2}-y+11}$

A) $\frac{y^{2}-3 y+12}{y^{2}+y+11}$
B) $\frac{y^{2}+12 y-3}{y^{2}-y-11}$
C) $\frac{y^{2}-y+11}{y^{2}+12 y+3}$
D) $\frac{y^{2}-y-11}{y^{2}+12 y-3}$

Question 39

Find the reciprocal.

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

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

Question 40

Divide and simplify.

- $\frac{5}{9} \div \frac{9}{4}$

A) $\frac{5}{4}$
B) $\frac{4}{5}$
C) $\frac{20}{81}$
D) $\frac{81}{20}$

Question 41

Divide and simplify.

- $\frac{3 x^{2}}{4} \div \frac{x^{3}}{16}$

A) $\frac{12}{x}$
B) $\frac{x}{12}$
C) $\frac{x^{5}}{21}$
D) $\frac{12 x^{2}}{x^{3}}$

Question 42

Divide and simplify.

- $\frac{7 p-7}{p} \div \frac{8 p-8}{5 p^{2}}$

A) $\frac{35 p}{8}$
B) $\frac{8}{35 p}$
C) $\frac{35 p^{3}-35 p^{2}}{8 p^{2}-8 p}$
D) $\frac{56 p^{2}+112 p+56}{5 p^{3}}$

Question 43

Divide and simplify.

- $\frac{10}{y-x} \div \frac{5}{x-y}$

A) 2
B) $\frac{2(x-y)}{y-x}$
C) -2
D) -10

Question 44

Divide and simplify.

- $\frac{-12+4 x}{12} \div \frac{21-7 x}{21}$

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

Question 45

Divide and simplify.

- $\frac{x^{2}-64}{x} \div \frac{3 x+24}{x-8}$

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

Question 46

Divide and simplify.

- $\frac{x^{2}+9 x+20}{x+4} \div \frac{x^{2}-25}{4 x-20}$

A) $\frac{1}{4}$
B) $\frac{4 x-20}{x-5}$
C) 4
D) $\frac{(x+5)^{2}}{4}$

Question 47

Divide and simplify.

- $\frac{z^{2}+12 z+35}{z^{2}+13 z+40} \div \frac{z^{2}+7 z}{z^{2}+18 z+80}$

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

Question 48

Divide and simplify.

- $\frac{z^{2}+12 z+35}{z^{2}+14 z+49} \div \frac{z^{2}+5 z}{z^{2}+4 z-21}$

A) $\frac{z-3}{z^{2}+7 z}$
B) $\frac{z}{z^{2}+14 z+49}$
C) $\frac{z-3}{z}$
D) $\mathrm{z}-3$

Question 49

Divide and simplify.

- $\frac{a^{2}+3 a-28}{a^{2}-14 a+49} \div \frac{a^{2}-8 a+16}{a^{2}+6 a-7}$

A) $-\frac{a-1}{a-4}$
B) $\frac{(a+7)^{2}(a-1)}{(a-7)^{2}(a-4)}$
C) $\frac{(a+7)^{2}(a-1)}{(a-7)^{2}}$
D) $\frac{(a-4)^{3}}{(a-7)^{2}(a-1)}$

Question 50

Find the LCM.

-20,15

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

Question 51

Find the LCM.

- $10,6,15$

A) 15
B) 10
C) 6
D) 30

Question 52

Find the LCM.

- $60,24,90$

A) 360
B) 120
C) 72
D) 180

Question 53

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

- $\frac{8}{9}+\frac{2}{7}$

A) $\frac{37}{8}$
B) $\frac{10}{63}$
C) $\frac{74}{63}$
D) $\frac{5}{8}$

Question 54

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

- $\frac{10}{13}+\frac{1}{8}$

A) $\frac{11}{21}$
B) $\frac{93}{104}$
C) $\frac{11}{104}$
D) $\frac{31}{7}$

Question 55

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

- $\frac{1}{24}+\frac{5}{54}$

A) $\frac{29}{13}$
B) $\frac{1}{216}$
C) $\frac{29}{216}$
D) $\frac{1}{13}$

Question 56

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

- $\frac{1}{4}+\frac{2}{7}+\frac{1}{3}$

A) $\frac{2}{7}$
B) $\frac{1}{21}$
C) $\frac{73}{14}$
D) $\frac{73}{84}$

Question 57

Find the LCM.

- $42 x, 6 x^{2}, 7 x^{3}$

A) $42 x^{2}$
B) $7 x^{3}$
C) $42 x^{5}$
D) $42 x^{3}$

Question 58

Find the LCM.

- $8 x y, 12 x^{2}$

A) $24 x^{2} y$
B) $24 x y^{3}$
C) $24 x y^{2}$
D) $24 x^{3} y$

Question 59

Find the LCM.

- $t, t-6$

A) 12
B) $\mathrm{t}$
C) $t-6$
D) $t(t-6)$

Question 60

Find the LCM.

- $4 a-12, a^{2}-3 a$

A) $4 a^{2}-3$
B) $4 \mathrm{a}(\mathrm{a}-3)$
C) $4 a^{2}-12$
D) $4 \mathrm{a}-3$

Question 61

Find the LCM.

- $n, 8+n, 8-n$

A) $64 n^{2}$
B) $n^{2}+64$
C) $n+8$
D) $n(8+n)(8-n)$

Question 62

Find the LCM.

- $r^{2}+10 r+25, r^{2}+5 r$

A) $r(r+1)(r+5)$
B) $r(r+5)$
C) $(r+5)^{2}$
D) $r(r+5)^{2}$

Question 63

Find the LCM.

- $m^{2}+4 m, m^{2}+7 m+12$

A) $m(m+1)^{2}$
B) $m(m+4)(m+3)$
C) $m(x+1)(m+3)$
D) $(m+1)^{2}$

Question 64

Find the LCM.

- $x^{2}-8 x+16, x^{2}-3 x-4$

A) $(x-4)(x+1)$
B) $(x+4)(x+4)(x+1)$
C) $(x-4)(x-4)$
D) $(x-4)(x-4)(x+1)$

Question 65

Find the LCM.

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

A) $-6(x-3)(x-1)$
B) $-6(x-3)(x+1)$
C) $-6(x+3)(x+1)$
D) $-6(x+3)(x-1)$

Question 66

Add. Simplify, if possible.

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

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

Question 67

Add. Simplify, if possible.

- $\frac{5}{13 x}+\frac{3}{13 x}$

A) $\frac{8}{26 x}$
B) 1
C) $\frac{8}{13 x}$
D) $\frac{13 x}{8}$

Question 68

Add. Simplify, if possible.

- $\frac{9}{8 x^{2}}+\frac{3}{8 x^{2}}$

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

Question 69

Add. Simplify, if possible.

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

A) $\frac{7 x+1}{2 x-3}$
B) $-2 x-\frac{7}{3}$
C) $\frac{-4 x+7}{4 x-6}$
D) $\frac{-4 x+7}{2 x-3}$

Question 70

Add. Simplify, if possible.

- $\frac{7}{x+y}+\frac{8}{y+x}$

A) $\frac{-1}{x+y}$
B) $\frac{15}{2 x+2 y}$
C) $\frac{15}{x+y}$
D) $\frac{7}{x}+\frac{8}{y}$

Question 71

Add. Simplify, if possible.

- $\frac{m^{2}-10 m}{m-6}+\frac{24}{m-6}$

A) $m-6$
B) $m+4$
C) $\frac{m^{2}-10 m+24}{m-6}$
D) $m-4$

Question 72

Add. Simplify, if possible.

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

A) $\frac{10}{x-3}$
B) $\frac{2}{x-3}$
C) $\frac{-2}{x-3}$
D) $\frac{24}{x-3}$

Question 73

Add. Simplify, if possible.

- $\frac{5}{2-m}+\frac{4}{m-2}$

A) $\frac{1}{2-m}$
B) $\frac{20}{2-m}$
C) $\frac{-1}{2-m}$
D) $\frac{9}{2-m}$

Question 74

Add. Simplify, if possible.

- $\frac{4}{m-n^{2}}+\frac{9}{n^{2}-m}$

A) $\frac{5}{m-n^{2}}$
B) $\frac{13}{m-n^{2}}$
C) $\frac{36}{m-n^{2}}$
D) $\frac{-5}{m-n^{2}}$

Question 75

Add. Simplify, if possible.

- $\frac{6}{9 x-7}+\frac{1}{7-9 x}$

A) $\frac{7}{9 x-7}$
B) $\frac{-5}{9 x-7}$
C) $\frac{5}{9 x-7}$
D) $\frac{-7}{9 x-7}$

Question 76

Add. Simplify, if possible.

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

A) $\frac{9 x+18}{(x-8)^{3}}$
B) $\frac{9 x+18}{x-8}$
C) $\frac{11 x+10}{x-8}$
D) $\frac{9 x+10}{x-8}$

Question 77

Add. Simplify, if possible.

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

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

Question 78

Add. Simplify, if possible.

- $\frac{3}{r}+\frac{4}{r-5}$

A) $\frac{15 r-7}{r(r-5)}$
B) $\frac{7 r-15}{r(5-r)}$
C) $\frac{15 r-7}{r(5-r)}$
D) $\frac{7 r-15}{r(r-5)}$

Question 79

Add. Simplify, if possible.

- $\frac{5}{r}+\frac{9}{r-7}$

A) $\frac{14 r-35}{r(7-r)}$
B) $\frac{14 \mathrm{r}-35}{\mathrm{r}(\mathrm{r}-7)}$
C) $\frac{35 r-14}{r(r-7)}$
D) $\frac{35 r-14}{r(7-r)}$

Question 80

Add. Simplify, if possible.

- $\frac{m+3}{m^{2}-4 m+3}+\frac{4 m-1}{m^{2}-5 m+6}$

A) $\frac{5 m+2}{2 m^{2}-9 m+9}$
B) $5 m+2$
C) $\frac{5 m^{2}-4 m-5}{(m-3)(m-1)(m-2)}$
D) $\frac{5 m^{2}-4 m-5}{(m+3)(m+1)(m+2)}$