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Given two rational expressions, divide them.
Divide the rational expressions and express the quotient in simplest form:
Divide the rational expressions and express the quotient in simplest form:
Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.
We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.
The easiest common denominator to use will be the least common denominator , or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were and then the LCD would be
Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of by and the expression with a denominator of by
Given two rational expressions, add or subtract them.
Add the rational expressions:
First, we have to find the LCD. In this case, the LCD will be We then multiply each expression by the appropriate form of 1 to obtain as the denominator for each fraction.
Now that the expressions have the same denominator, we simply add the numerators to find the sum.
Subtract the rational expressions:
Do we have to use the LCD to add or subtract rational expressions?
No. Any common denominator will work, but it is easiest to use the LCD.
A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression can be simplified by rewriting the numerator as the fraction and combining the expressions in the denominator as We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get which is equal to
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