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Simplifying a Complex Rational Expression by Using the LCD

Simplifying 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 this example from the previous lesson. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by \(\frac{\text{LCD}}{\text{LCD}}\) we are multiplying by 1, so the value stays the same.

Example

Simplify: \(\cfrac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

Solution

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

Example: How to Simplify a Complex Rational Expression by Using the LCD

Simplify: \(\cfrac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)

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.

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.

Example

Simplify: \(\cfrac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}.\)

Solution

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

Example

Simplify: \(\cfrac{\frac{4}{{m}^{2}-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}.\)

Solution

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

Example

Simplify: \(\cfrac{\frac{y}{y+1}}{1+\frac{1}{y-1}}.\)

Solution

 .
Find the LCD of all fractions in the complexrational expression. 
The LCD is \(\left(y+1\right)\left(y-1\right)\). 
Multiply the numerator and denominator by the LCD..
Distribute in the denominator and simplify..
Simplify..
Simplify the denominator, and leave the numeratorfactored..
 .
Factor the denominator, and remove factors commonwith the numerator..
Simplify..

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