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7.3 Simplify Complex Rational Expressions

Fractions stacked inside fractions, cleared either by rewriting the main bar as division or by multiplying every term by the overall LCD.

Intermediate Algebra 2e, 7.3 Simplify Complex Rational Expressions

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45 exercises

7.3 Exercise 1

Before you get started, take this readiness quiz.

Simplify: 35910.\frac{\frac{3}{5}}{\frac{9}{10}} .
If you missed this problem, review the reference.

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7.3 Exercise 2

Simplify: 1−1342+4·5.\frac{1 - \frac{1}{3}}{4^{2} + 4 \cdot 5} .
If you missed this problem, review the reference.

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7.3 Exercise 3

Solve: 12x+14=18.\frac{1}{2 x} + \frac{1}{4} = \frac{1}{8} .
If you missed this problem, review the reference.

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7.3 Exercise 4

Simplify the complex rational expression by writing it as division: 6x−43x2−16.\frac{\frac{6}{x - 4}}{\frac{3}{x^{2} - 16}} .

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7.3 Exercise 5

Simplify the complex rational expression by writing it as division: 2x2−13x+1.\frac{\frac{2}{x^{2} - 1}}{\frac{3}{x + 1}} .

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7.3 Exercise 6

Simplify the complex rational expression by writing it as division: 1x2−7x+122x−4.\frac{\frac{1}{x^{2} - 7 x + 12}}{\frac{2}{x - 4}} .

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7.3 Exercise 8

Simplify the complex rational expression by writing it as division: 12+2356+112.\frac{\frac{1}{2} + \frac{2}{3}}{\frac{5}{6} + \frac{1}{12}} .

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7.3 Exercise 9

Simplify the complex rational expression by writing it as division: 34−1318+56.\frac{\frac{3}{4} - \frac{1}{3}}{\frac{1}{8} + \frac{5}{6}} .

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7.3 Exercise 11

Simplify the complex rational expression by writing it as division: 1x+1y1x−1y.\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}} .

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7.3 Exercise 12

Simplify the complex rational expression by writing it as division: 1a+1b1a2−1b2\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^{2}} - \frac{1}{b^{2}}}.

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7.3 Exercise 14

Simplify the complex rational expression by writing it as division: b−3bb+52b+5+1b−5.\frac{b - \frac{3 b}{b + 5}}{\frac{2}{b + 5} + \frac{1}{b - 5}} .

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7.3 Exercise 15

Simplify the complex rational expression by writing it as division: 1−3c+41c+4+c3.\frac{1 - \frac{3}{c + 4}}{\frac{1}{c + 4} + \frac{c}{3}} .

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7.3 Exercise 17

Simplify the complex rational expression by using the LCD: 12+15110+15.\frac{\frac{1}{2} + \frac{1}{5}}{\frac{1}{10} + \frac{1}{5}} .

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7.3 Exercise 18

Simplify the complex rational expression by using the LCD: 14+3812−516.\frac{\frac{1}{4} + \frac{3}{8}}{\frac{1}{2} - \frac{5}{16}} .

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7.3 Exercise 20

Simplify the complex rational expression by using the LCD: 1a+1bab+ba.\frac{\frac{1}{a} + \frac{1}{b}}{\frac{a}{b} + \frac{b}{a}} .

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7.3 Exercise 21

Simplify the complex rational expression by using the LCD: 1x2−1y21x+1y.\frac{\frac{1}{x^{2}} - \frac{1}{y^{2}}}{\frac{1}{x} + \frac{1}{y}} .

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7.3 Exercise 23

Simplify the complex rational expression by using the LCD: 3x+25x−2−3x2−4.\frac{\frac{3}{x + 2}}{\frac{5}{x - 2} - \frac{3}{x^{2} - 4}} .

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7.3 Exercise 24

Simplify the complex rational expression by using the LCD: 2x−7−1x+76x+7−1x2−49.\frac{\frac{2}{x - 7} - \frac{1}{x + 7}}{\frac{6}{x + 7} - \frac{1}{x^{2} - 49}} .

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7.3 Exercise 26

Simplify the complex rational expression by using the LCD: 3x2+7x+104x+2+1x+5.\frac{\frac{3}{x^{2} + 7 x + 10}}{\frac{4}{x + 2} + \frac{1}{x + 5}} .

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7.3 Exercise 27

Simplify the complex rational expression by using the LCD: 4yy+5+2y+63yy2+11y+30.\frac{\frac{4 y}{y + 5} + \frac{2}{y + 6}}{\frac{3 y}{y^{2} + 11 y + 30}} .

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7.3 Exercise 29

Simplify the complex rational expression by using the LCD: xx+31+1x+3.\frac{\frac{x}{x + 3}}{1 + \frac{1}{x + 3}} .

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7.3 Exercise 30

Simplify the complex rational expression by using the LCD: 1+1x−13x+1.\frac{1 + \frac{1}{x - 1}}{\frac{3}{x + 1}} .

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7.3 Exercise 31

Simplify each complex rational expression by writing it as division.

2aa+44a2a2−16\frac{\frac{2 a}{a + 4}}{\frac{4 a^{2}}{a^{2} - 16}}

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7.3 Exercise 33

Simplify each complex rational expression by writing it as division.

5c2+5c−1410c+7\frac{\frac{5}{c^{2} + 5 c - 14}}{\frac{10}{c + 7}}

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7.3 Exercise 35

Simplify each complex rational expression by writing it as division.

12+5623+79\frac{\frac{1}{2} + \frac{5}{6}}{\frac{2}{3} + \frac{7}{9}}

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7.3 Exercise 37

Simplify each complex rational expression by writing it as division.

23−1934+56\frac{\frac{2}{3} - \frac{1}{9}}{\frac{3}{4} + \frac{5}{6}}

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7.3 Exercise 39

Simplify each complex rational expression by writing it as division.

nm+1n1n−nm\frac{\frac{n}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m}}

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7.3 Exercise 41

Simplify each complex rational expression by writing it as division.

1r+1t1r2−1t2\frac{\frac{1}{r} + \frac{1}{t}}{\frac{1}{r^{2}} - \frac{1}{t^{2}}}

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7.3 Exercise 43

Simplify each complex rational expression by writing it as division.

x−2xx+31x+3+1x−3\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}

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7.3 Exercise 45

Simplify each complex rational expression by writing it as division.

2−2a+31a+3+a2\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}

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7.3 Exercise 47

Simplify each complex rational expression by using the LCD.

13+1814+112\frac{\frac{1}{3} + \frac{1}{8}}{\frac{1}{4} + \frac{1}{12}}

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7.3 Exercise 49

Simplify each complex rational expression by using the LCD.

56+29718−13\frac{\frac{5}{6} + \frac{2}{9}}{\frac{7}{18} - \frac{1}{3}}

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7.3 Exercise 51

Simplify each complex rational expression by using the LCD.

cd+1d1d−dc\frac{\frac{c}{d} + \frac{1}{d}}{\frac{1}{d} - \frac{d}{c}}

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7.3 Exercise 53

Simplify each complex rational expression by using the LCD.

1p+1q1p2−1q2\frac{\frac{1}{p} + \frac{1}{q}}{\frac{1}{p^{2}} - \frac{1}{q^{2}}}

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7.3 Exercise 55

Simplify each complex rational expression by using the LCD.

2x+53x−5+1x2−25\frac{\frac{2}{x + 5}}{\frac{3}{x - 5} + \frac{1}{x^{2} - 25}}

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7.3 Exercise 57

Simplify each complex rational expression by using the LCD.

5z2−64+3z+81z+8+2z−8\frac{\frac{5}{z^{2} - 64} + \frac{3}{z + 8}}{\frac{1}{z + 8} + \frac{2}{z - 8}}

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7.3 Exercise 59

Simplify each complex rational expression by using the LCD.

4a2−2a−151a−5+2a+3\frac{\frac{4}{a^{2} - 2 a - 15}}{\frac{1}{a - 5} + \frac{2}{a + 3}}

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7.3 Exercise 61

Simplify each complex rational expression by using the LCD.

5c+2−3c+75cc2+9c+14\frac{\frac{5}{c + 2} - \frac{3}{c + 7}}{\frac{5 c}{c^{2} + 9 c + 14}}

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7.3 Exercise 63

Simplify each complex rational expression by using the LCD.

2+1p−35p−3\frac{2 + \frac{1}{p - 3}}{\frac{5}{p - 3}}

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7.3 Exercise 65

Simplify each complex rational expression by using the LCD.

mm+54+1m−5\frac{\frac{m}{m + 5}}{4 + \frac{1}{m - 5}}

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7.3 Exercise 67

Simplify each complex rational expression using either method.

34−2712+514\frac{\frac{3}{4} - \frac{2}{7}}{\frac{1}{2} + \frac{5}{14}}

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7.3 Exercise 69

Simplify each complex rational expression using either method.

2a+41a2−16\frac{\frac{2}{a + 4}}{\frac{1}{a^{2} - 16}}

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7.3 Exercise 71

Simplify each complex rational expression using either method.

3m+3n1m2−1n2\frac{\frac{3}{m} + \frac{3}{n}}{\frac{1}{m^{2}} - \frac{1}{n^{2}}}

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7.3 Exercise 73

Simplify each complex rational expression using either method.

x−3xx+23x+2+3x−2\frac{x - \frac{3 x}{x + 2}}{\frac{3}{x + 2} + \frac{3}{x - 2}}

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7.3 Exercise 75

In this section, you learned to simplify the complex fraction 3x+2xx2−4\frac{\frac{3}{x + 2}}{\frac{x}{x^{2} - 4}} two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

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Adapted from Intermediate Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Intermediate Algebra 2e. Imported at revision 77a933c.