﻿ Complex Rational Expressions Worksheet - Page 3 | Problems & Solutions

# Complex Rational Expressions Worksheet - Page 3

Complex Rational Expressions Worksheet
• Page 3
21.
Simplify:
 a. $\frac{10\left(x³+20x²+109x+90\right)}{10x³+199x²+980x-11}$ b. $\frac{10\left(x³+20x²+109x+90\right)}{10x³+199x²+69x+11}$ c. $\frac{9\left(x³+20x²\right)}{199x²+980x-11}$ d. $\frac{9\left(109x+90\right)}{10x³+199x²}$

#### Solution:

x+9x+90x+1010x-1+89(x+10)²

= (x+10)²(x+10)² · x+9x+90x+1010x-1+89(x+10)²
[The LCD is (x + 10)2, so multiply numerator and denominator by (x + 10)2.]

= x(x+10)²+9x(x+10)²+90(x+10)10x(x+10)²-(x+10)²+89

= x(x²+20x+100)+9x(x²+20x+100)+90(x+10)10x(x²+20x+100) - (x²+20x+100)+89
[Simplify.]

= 10x³+200x²+1090x+90010x³+199x²+980x-11
[Distributive property.]

= 10(x³+20x²+109x+90)10x³+199x²+980x-11

22.
Simplify: $\frac{\frac{7a}{{\left(a+3\right)}^{3}}+\frac{1}{\left(a+3\right)}}{\frac{11}{{\left(a+3\right)}^{2}}+\frac{9}{{\left(a+3\right)}^{2}}}$
 a. $\frac{a+7}{20a+60}$ b. $\frac{a+3}{11a+33}$ c. $\frac{a²+13a+9}{9a+27}$ d. $\frac{a²+13a+9}{20a+60}$

#### Solution:

7a(a+3)3+1(a+3)11(a+3)2+9(a+3)2

= (a+3)³(a+3)³(7a(a+3)³+1(a+3)) / (11(a+3)²+9(a+3)²)
[The LCD is (a + 3)3, so multiply numerator and denominator by (a + 3)3.]

= [7a(a+3)³(a+3)³+1a+3(a+3)³] / [11(a+3)²(a+3)³+9(a+3)²(a+3)³]
[Distributive property.]

= 7a+(a+3)²11(a+3)+9(a+3)
[Simplify.]

= 7a+a²+6a+911a+33+9a+27
[Write (a + 3)2 as a2 + 6a + 9.]

= a²+13a+920a+60

23.
Simplify: $\frac{\frac{9}{a-7}}{1-\frac{8}{a-7}}$
 a. b. c. d.

#### Solution:

9a-71-8a-7

= a-7a-79a-71-8a-7
[The LCD is a - 7, so multiply by a - 7.]

= 9a-7(a-7)(a-7)-8a-7(a-7)
[Distributive property.]

= 9a-15
[Simplify.]

24.
Simplify:
$\frac{1+\frac{8}{a}+\frac{7}{a²}}{\frac{7}{a}+35}$
 a. $\frac{a+1}{1+5a}$ b. $\frac{7a\left(1+5a\right)}{\left(a+1\right)\left(a+7\right)}$ c. $\frac{\left(a+1\right)\left(a+7\right)}{7a\left(1+5a\right)}$ d. $\frac{\left(a+1\right)\left(a+7\right)}{\left(7+5a\right)}$

#### Solution:

1+8a+7a²7a+35

= a²a²1+8a+7a²7a+35
[The LCD is a2, so multiply by a2.]

= a²+8a+77a+35a²
[Distributive property.]

= (a+1)(a+7)7a(1+5a)
[Factor.]

25.
Simplify: $\frac{\frac{11}{4b-24}-\frac{5}{b+6}}{\frac{5}{5b-30}+\frac{6}{b+6}}$
 a. $\frac{-9b+186}{28b-120}$ b. $\frac{-9b-186}{28b+120}$ c. $\frac{9b+186}{28b-120}$ d. $\frac{9b-186}{28b-120}$

#### Solution:

114b-24-5b+655b-30+6b+6

= 114(b-6)-5b+655(b-6)+6b+6

11(b+6)-5×4(b-6)4(b-6)(b+6)5(b+6)+6×5(b-6)5(b-6)(b+6)

11(b+6)-20(b-6)5(b+6)+30(b-6) · 5 / 4
[Cancel common factors.]

= 55(b+6)-100(b-6)20(b+6)+120(b-6)
[Distributive property.]

= -45b+930140b-600
[Simplify.]

= -9b+18628b-120
[Simplify.]

26.
Simplify:
$\frac{\frac{100}{9x-99}-\frac{9}{x+11}}{\frac{95}{10x-110}+\frac{9}{x+11}}$
 a. $\frac{9\left(37x+11\right)}{2\left(19x+1991\right)}$ b. $\frac{9}{2}$ c. $\frac{9\left(19x+1991\right)}{2\left(37x+11\right)}$ d. $\frac{2\left(19x+1991\right)}{9\left(37x+11\right)}$

#### Solution:

1009x-99-9x+119510x-110+9x+11

= 100(x+11)-9[9(x-11)]9(x-11)(x+11)95(x+11)+9[10(x-11)]10(x-11)(x+11)
[Take LCD of numerator and denominator.]

= 100x+1100-81x+8919(x-11)(x+11)95x+1045+90x-99010(x-11)(x+11)
[Simplify.]

= 19x+19919(x-11)(x+11)185x+5510(x-11)(x+11)
[Simplify.]

= 19x+19919(x-11)(x+11) ÷185x+5510(x-11)(x+11)
[Rewrite.]

= 19x+19919(x-11)(x+11) 10(x-11)(x+11)185x+55
[Multiply by the reciprocal.]

= 10(19x+1991)9(185x+55) = 10*1(19x+1991)9*5(37x+11)
[Divide out the common factor.]

= 2(19x+1991)9(37x+11)
[Divide out the common factor.]