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Simplify: n 4 n n + 5 1 n + 5 + 1 n 5 .

Solution

.
Simplify the numerator and denominator.
Find the LCD and add the fractions in the numerator.
Find the LCD and add the fractions in the denominator.
.
Simplify the numerators. .
Subtract the rational expressions in the numerator and add in the denominator.

Simplify.
.
Rewrite as fraction division. .
Multiply the first times the reciprocal of the second. .
Factor any expressions if possible. .
Remove common factors. .
Simplify. .

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Simplify: b 3 b b + 5 2 b + 5 + 1 b 5 .

b ( b + 2 )

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Simplify: 1 3 c + 4 1 c + 4 + c 3 .

3 c + 3

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Simplify a complex rational expression by using the lcd

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in [link] . We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by LCD LCD we are multiplying by 1, so the value stays the same.

Simplify: 1 3 + 1 6 1 2 1 3 .

Solution

.
The LCD of all the fractions in the whole expression is 6.
Clear the fractions by multiplying the numerator and denominator by that LCD. .
Distribute. .
Simplify. .
.
.

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Simplify: 1 2 + 1 5 1 10 + 1 5 .

10 3

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Simplify: 1 4 + 3 8 1 2 5 16 .

7 3

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How to simplify a complex rational expression by using the lcd

Simplify: 1 x + 1 y x y y x .

Solution

The above image has 3 columns. It shows the steps on how to simplify a complex rational expression using the LCD for 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Step one is to find the LCD of all fractions in the complex rational expression. The LCD of all the fractions is x y. Multiply the numerator and denominator by the LCD. Step two is to multiply both the numerator and denominator by x y to get x y times 1 divided by x plus 1 divided by y divided x y times x divided by y minus y divided by x. Step three is to simplify the expression. Distribute to get x y times 1 divided by x plus x y times 1 divided y divided by x y times x divided by y minus x y times y divided by x. Simplify to get y plus x divided by x squared minus y squared. Remove common factors. Cross out y plus x in the numerator. Cross out x plus y in the numerator. Simplify to get 1 divided by x minus y.
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Simplify: 1 a + 1 b a b + b a .

b + a a 2 + b 2

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Simplify: 1 x 2 1 y 2 1 x + 1 y .

y x x y

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Simplify a complex rational expression by using the lcd.

  1. Find the LCD of all fractions in the complex rational expression.
  2. Multiply the numerator and denominator by the LCD.
  3. Simplify the expression.

Be sure to start by factoring all the denominators so you can find the LCD.

Simplify: 2 x + 6 4 x 6 4 x 2 36 .

Solution

.
Find the LCD of all fractions in the complex rational expression. The LCD is ( x + 6 ) ( x 6 ) .
Multiply the numerator and denominator by the LCD. .
Simplify the expression.
Distribute in the denominator. .
Simplify. .
Simplify. .
To simplify the denominator, distribute and combine like terms. .
Remove common factors. .
Simplify. .
Notice that there are no more factors common to the numerator and denominator.

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Simplify: 3 x + 2 5 x 2 3 x 2 4 .

3 x 6 5 x + 7

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Simplify: 2 x 7 1 x + 7 6 x + 7 1 x 2 49 .

x + 21 6 x + 43

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Simplify: 4 m 2 7 m + 12 3 m 3 2 m 4 .

Solution

.
Find the LCD of all fractions in the complex rational expression. The LCD is ( m 3 ) ( m 4 ) .
Multiply the numerator and denominator by the LCD. .
Simplify. .
Simplify. .
Distribute. .
Combine like terms. .

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Simplify: 3 x 2 + 7 x + 10 4 x + 2 + 1 x + 5 .

3 5 x + 22

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Simplify: 4 y y + 5 + 2 y + 6 3 y y 2 + 11 y + 30 .

6 y + 34 3 y

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Simplify: y y + 1 1 + 1 y 1 .

Solution

.
Find the LCD of all fractions in the complex rational expression.
The LCD is ( y + 1 ) ( y 1 ) .
Multiply the numerator and denominator by the LCD. .
Distribute in the denominator and simplify. .
Simplify. .
Simplify the denominator, and leave the numerator factored. .
.
Factor the denominator, and remove factors common with the numerator. .
Simplify. .

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Simplify: x x + 3 1 + 1 x + 3 .

x x + 4

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Simplify: 1 + 1 x 1 3 x + 1 .

x ( x + 1 ) 3 ( x 1 )

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Key concepts

  • To Simplify a Rational Expression by Writing it as Division
    1. Simplify the numerator and denominator.
    2. Rewrite the complex rational expression as a division problem.
    3. Divide the expressions.
  • To Simplify a Complex Rational Expression by Using the LCD
    1. Find the LCD of all fractions in the complex rational expression.
    2. Multiply the numerator and denominator by the LCD.
    3. Simplify the expression.

Practice makes perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

2 a a + 4 4 a 2 a 2 16

a 4 2

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3 b b 5 b 2 b 2 25

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5 c 2 + 5 c 14 10 c + 7

3 2 ( c 2 )

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8 d 2 + 9 d + 18 12 d + 6

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1 2 + 5 6 2 3 + 7 9

24 26

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2 3 1 9 3 4 + 5 6

20 57

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n m + 1 n 1 n n m

n 2 + m m n 2

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1 r + 1 t 1 r 2 1 t 2

r t t r

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2 v + 2 w 1 v 2 1 w 2

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x 2 x x + 3 1 x + 3 + 1 x 3

( x + 1 ) ( x 3 ) 2

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y 2 y y 4 2 y 4 2 y + 4

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2 2 a + 3 1 a + 3 + a 2

4 a + 1

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4 4 b 5 1 b 5 + b 4

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Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

1 3 + 1 8 1 4 + 1 12

11 8

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5 6 + 2 9 7 18 1 3

19

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c d + 1 d 1 d d c

c 2 + c c d

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1 p + 1 q 1 p 2 1 q 2

p q q p

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2 r + 2 t 1 r 2 1 t 2

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2 x + 5 3 x 5 + 1 x 2 25

2 x 10 3 x + 16

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5 y 4 3 y + 4 + 2 y 2 16

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5 z 2 64 + 3 z + 8 1 z + 8 + 2 z 8

3 z 19 3 z + 8

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3 s + 6 + 5 s 6 1 s 2 36 + 4 s + 6

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4 a 2 2 a 15 1 a 5 + 2 a + 3

4 3 a 2

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5 b 2 6 b 27 3 b 9 + 1 b + 3

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5 c + 2 3 c + 7 5 c c 2 + 9 c + 14

2 c + 29 5 c

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6 d 4 2 d + 7 2 d d 2 + 3 d 28

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2 + 1 p 3 5 p 3

( 2 p 5 ) 5

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m m + 5 4 + 1 m 5

m ( m 5 ) 4 m 2 + m 95

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Simplify

In the following exercises, use either method.

3 4 2 7 1 2 + 5 14

13 24

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2 a + 4 1 a 2 16

2 ( a 4 )

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3 b 2 3 b 40 5 b + 5 2 b 8

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3 m + 3 n 1 m 2 1 n 2

3 m n n m

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2 r 9 1 r + 9 + 3 r 2 81

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x 3 x x + 2 3 x + 2 + 3 x 2

( x 1 ) ( x 2 ) 6

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Everyday math

Electronics The resistance of a circuit formed by connecting two resistors in parallel is 1 1 R 1 + 1 R 2 .

  1. Simplify the complex fraction 1 1 R 1 + 1 R 2 .
  2. Find the resistance of the circuit when R 1 = 8 and R 2 = 12 .

R 1 R 2 R 2 + R 1
24 5

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Ironing Lenore can do the ironing for her family’s business in h hours. Her daughter would take h + 2 hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is 1 1 h + 1 h + 2 .

  1. Simplify the complex fraction 1 1 h + 1 h + 2 .
  2. Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if h = 4 .
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Writing exercises

In this section, you learned to simplify the complex fraction 3 x + 2 x x 2 4 two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?

Answers will vary.

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Efraim wants to start simplifying the complex fraction 1 a + 1 b 1 a 1 b by cancelling the variables from the numerator and denominator. Explain what is wrong with Efraim’s plan.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.

After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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