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Given a complex rational expression, simplify it.

  1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.
  2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.
  3. Rewrite as the numerator divided by the denominator.
  4. Rewrite as multiplication.
  5. Multiply.
  6. Simplify.

Simplifying complex rational expressions

Simplify: y + 1 x x y .

Begin by combining the expressions in the numerator into one expression.

y x x + 1 x    Multiply by  x x  to get LCD as denominator . x y x + 1 x x y + 1 x    Add numerators .

Now the numerator is a single rational expression and the denominator is a single rational expression.

x y + 1 x x y

We can rewrite this as division, and then multiplication.

x y + 1 x ÷ x y x y + 1 x y x Rewrite as multiplication . y ( x y + 1 ) x 2 Multiply .
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Simplify: x y y x y

x 2 y 2 x y 2

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Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

Access these online resources for additional instruction and practice with rational expressions.

Key concepts

  • Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See [link] .
  • We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See [link] .
  • To divide rational expressions, multiply by the reciprocal of the second expression. See [link] .
  • Adding or subtracting rational expressions requires finding a common denominator. See [link] and [link] .
  • Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See [link] .

Section exercises

Verbal

How can you use factoring to simplify rational expressions?

You can factor the numerator and denominator to see if any of the terms can cancel one another out.

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How do you use the LCD to combine two rational expressions?

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Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.

True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

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Algebraic

For the following exercises, simplify the rational expressions.

x 2 16 x 2 5 x + 4

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y 2 + 10 y + 25 y 2 + 11 y + 30

y + 5 y + 6

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6 a 2 24 a + 24 6 a 2 24

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9 b 2 + 18 b + 9 3 b + 3

3 b + 3

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2 x 2 + 7 x 4 4 x 2 + 2 x 2

x + 4 2 x + 2

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6 x 2 + 5 x 4 3 x 2 + 19 x + 20

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a 2 + 9 a + 18 a 2 + 3 a 18

a + 3 a 3

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3 c 2 + 25 c 18 3 c 2 23 c + 14

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12 n 2 29 n 8 28 n 2 5 n 3

3 n 8 7 n 3

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For the following exercises, multiply the rational expressions and express the product in simplest form.

x 2 x 6 2 x 2 + x 6 2 x 2 + 7 x 15 x 2 9

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c 2 + 2 c 24 c 2 + 12 c + 36 c 2 10 c + 24 c 2 8 c + 16

c 6 c + 6

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2 d 2 + 9 d 35 d 2 + 10 d + 21 3 d 2 + 2 d 21 3 d 2 + 14 d 49

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10 h 2 9 h 9 2 h 2 19 h + 24 h 2 16 h + 64 5 h 2 37 h 24

1

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6 b 2 + 13 b + 6 4 b 2 9 6 b 2 + 31 b 30 18 b 2 3 b 10

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2 d 2 + 15 d + 25 4 d 2 25 2 d 2 15 d + 25 25 d 2 1

d 2 25 25 d 2 1

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6 x 2 5 x 50 15 x 2 44 x 20 20 x 2 7 x 6 2 x 2 + 9 x + 10

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t 2 1 t 2 + 4 t + 3 t 2 + 2 t 15 t 2 4 t + 3

t + 5 t + 3

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2 n 2 n 15 6 n 2 + 13 n 5 12 n 2 13 n + 3 4 n 2 15 n + 9

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36 x 2 25 6 x 2 + 65 x + 50 3 x 2 + 32 x + 20 18 x 2 + 27 x + 10

6 x 5 6 x + 5

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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