﻿ Complex Rational Expressions Worksheet | Problems & Solutions

# Complex Rational Expressions Worksheet

Complex Rational Expressions Worksheet
• Page 1
1.
Simplify:
$\frac{\frac{6x}{x²+4x+4}}{\frac{5}{x+2}+\frac{6}{x+2}}$
 a. $\frac{6x}{11x+22}$ b. $\frac{6x}{11x+2}$ c. $\frac{11x+22}{6x}$ d. $\frac{6x}{11x-2}$

#### Solution:

6xx²+4x+45x+2+6x+2

6xx²+4x+45x+2+6x+2 = 6x(x+2)(x+2)5x+2+6x+2
[Factor x2 + 4x + 4.]

= 6x(x+2)(x+2)5(x+2)+6(x+2)(x+2)(x+2)
[Take LCD of (x + 2) and (x + 2).]

= 6x(x+2)(x+2)11x+22(x+2)(x+2)
[Simplify.]

= 6x(x+2)(x+2) ÷11x+22(x+2)(x+2)
[Rewrite.]

= 6x(x+2)(x+2) (x+2)(x+2)11x+22
[Multiply by the reciprocal.]

= 6x11x+22
[Divide out the common factor.]

2.
Simplify:
$\frac{\frac{36}{5x-35}-\frac{5}{x+7}}{\frac{33}{6x-42}+\frac{5}{x+7}}$
 a. $\frac{2\left(11x+427\right)}{35\left(3x+1\right)}$ b. $\frac{35}{2}$ c. $\frac{35\left(11x+427\right)}{2\left(3x+1\right)}$ d. $\frac{35\left(3x+1\right)}{2\left(11x+427\right)}$

#### Solution:

365x-35-5x+7336x-42+5x+7

= 36(x+7)-5[5(x-7)]5(x-7)(x+7)33(x+7)+5[6(x-7)]6(x-7)(x+7)
[Take LCD of numerator and denominator.]

= 36x+252-25x+1755(x-7)(x+7)33x+231+30x-2106(x-7)(x+7)
[Simplify.]

= 11x+4275(x-7)(x+7)63x+216(x-7)(x+7)
[Simplify.]

= 11x+4275(x-7)(x+7) ÷63x+216(x-7)(x+7)
[Rewrite.]

= 11x+4275(x-7)(x+7) 6(x-7)(x+7)63x+21
[Multiply by the reciprocal.]

= 6(11x+427)5(63x+21)
[Divide out the common factor.]

= 2(11x+427)35(3x+1)
[Divide out the common factor.]

3.
Simplify:
$\frac{\frac{4x}{x²+8x+16}}{\frac{3}{x+4}+\frac{2x}{\left(x+4\right)²}}$
 a. b. c. d.

#### Solution:

4xx²+8x+163x+4+2x(x+4)²

= 4x(x+4)²3x+4+2x(x+4)²
[Factor.]

= 4x(x+4)²3(x+4)+2x(x+4)²
[Take LCD of the denominator.]

= 4x(x+4)² ÷5x+12(x+4)²
[Simplify and rewrite.]

= 4x(x+4)² (x+4)²5x+12
[Multiply by the reciprocal.]

= 4x5x + 12
[Multiply with the reciprocal.]

4.
Simplify:
- $\frac{3}{\left[\left(b+\frac{12}{a}\right)\right]}$
 a. - b. - c. d. -

#### Solution:

- 3[b + (12a)]

= - 3(ab + 12a)
[Take LCD and simplify the denominator.]

= - 3 ÷ ab + 12a
[Rewrite.]

= - 3 · aab + 12
[Multiply with the reciprocal.]

= - 3aab + 12

5.
Simplify:
$\frac{8}{5+\frac{x}{y}}$
 a. $\frac{8}{5y+x}$ b. $\frac{8y}{5y+xy}$ c. $\frac{8}{5y+xy}$ d. $\frac{8y}{5y+x}$

#### Solution:

85+xy

= yy85+xy
[The LCD is y, so multiply by y.]

= 8y5y+xyy
[Distributive property.]

= 8y5y+x

6.
Simplify:
$\frac{\left(10+x+\frac{10}{x-11}\right)}{\left(11-x-\frac{90}{9x-99}\right)}$
 a. - $\frac{x²-x-100}{9x²-198x+1179}$ b. $\frac{x²+21x+100}{x²-22x+131}$ c. $\frac{x²-x-100}{x²-22x+131}$ d. - $\frac{x²-x-100}{x²-22x+131}$

#### Solution:

(10+x+10x-11)(11-x-909x-99)

= 10(x-11)+x(x-11)+10x-1111(9x-99)-x(9x-99)-909x-99
[Take LCD in the numerator and the denominator.]

= (10x-110+x²-11x+10x-11)(99x-1089-9x²+99x-909x-99)
[Simplify.]

= x²-x-100x-11-(x²-22x+131)(x-11)
[Simplify and divide out the common factor 9.]

= x²-x-100x-11 ÷-[x²-22x+131]x-11
[Take out the common factor.]

= x²-x-100x-11 -(x-11)[x²-22x+131]
[Cancel the common factors.]

= - x²-x-100x²-22x+131
[Divide out the common factor (x - 11).]

7.
Simplify:
$\frac{\frac{-3}{c}+4+\frac{4}{c-2}}{-c-\frac{5}{c}+\frac{2}{\left(c-2\right)²}}$
 a. $\frac{4c³+15c²+20c}{{c}^{4}+4c³-9c²}$ b. $\frac{4c³-15c²+20c-12}{-{c}^{4}+4c³-9c²+22c-20}$ c. $\frac{-15c²+20c-12}{{c}^{4}-4c³+9c²-22c+20}$ d. None of the above

#### Solution:

-3c+4+4c-2-c-5c+2(c-2)²

= -3(c-2)+4c(c-2)+4cc(c-2)- c[c(c-2)²]-5(c-2)²+2cc(c-2)²
[Take LCD in the numerator and the denominator.]

= 4c²-7c+6-c4+4c³-9c²+22c-20(c-2)
[Simplify.]

= [4c²-7c+6](c-2)-c4+4c³-9c²+22c-20
[Divide out the common factor c(c - 2).]

= 4c³-15c²+20c-12-c4+4c³-9c²+22c-20
[Multiply with the reciprocal.]

8.
Simplify:
$\frac{\frac{5}{x}+\frac{9}{y}}{\frac{-4}{x}+\frac{6}{y}}$
 a. b. c. d. $\frac{5y+9x}{-4y+6x}$

#### Solution:

5x+9y-4x+6y

= xyxy5x+9y-4x+6y
[The LCD is xy, so multiply by xyxy.]

= xy5x+xy9yxy-4x+xy6y
[Distributive property.]

= 5y+9x-4y+6x

9.
Simplify:

 a. b. c. d.

#### Solution:

5m(10a-10b)2 +6n(a-b)310l(10a-10b)3

= 5m102(a-b)2+6n(a-b)310l103(a-b)3
[Take out the common factor.]

= m20(a-b)2+6n(a-b)3l100(a-b)3
[Cancel the common factors.]

= m(a-b)+120n20(a-b)3l100(a-b)3
[Take LCD in the numerator.]

= ma-mb+120n20(a-b)3l100(a-b)3
[Simplify.]

= ma-mb+120n20(a-b)3÷l100(a-b)3
[Rewrite.]

= ma-mb+120n20(a-b)3100(a-b)3l
[Multiply with the reciprocal.]

= 5ma - 5mb + 600nl
[Cancel the common factors.]

10.
Simplify:
$\left(\frac{13}{c}\right)/\left(\frac{12}{c}\right)$
 a. $\frac{156}{c²}$ b. $\frac{13}{12}$ c. $\frac{12}{13}$ d. $\frac{12c²}{13}$

#### Solution:

(13c)/(12c)

= 13c ÷12c
[Rewrite.]

= 13c c12
[Multiply with the reciprocal.]

= 13 / 12
[Cancel the common factors.]