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Adding Rational Expressions With Different Denominators

Adding Rational Expressions With Different Denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.

Example

Add: $$\cfrac{7}{12}+\cfrac{5}{18}.$$

Solution  Find the LCD of 12 and 18.  Rewrite each fraction as an equivalent fraction with the LCD.  Add the fractions.  The fraction cannot be simplified.  Now we will add rational expressions whose denominators are monomials.

Example

Add: $$\cfrac{5}{12{x}^{2}y}+\cfrac{4}{21x{y}^{2}}.$$

Solution  Find the LCD of $$12{x}^{2}y$$ and $$21x{y}^{2}$$.    Rewrite each rational expression as an equivalent fraction with the LCD.  Simplify.  Add the rational expressions.  There are no factors common to the numerator and denominator. The fraction cannot be simplified.

Now we are ready to tackle polynomial denominators.

Example: How to Add Rational Expressions with Different Denominators

Add: $$\cfrac{3}{x-3}+\cfrac{2}{x-2}.$$

Solution   The steps to use to add rational expressions are summarized in the following procedure box.

1. Determine if the expressions have a common denominator.
Yes – go to step 2.
No – Rewrite each rational expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.
2. Add the rational expressions.
3. Simplify, if possible.

Example

Add: $$\cfrac{2a}{2ab+{b}^{2}}+\cfrac{3a}{4{a}^{2}-{b}^{2}}.$$

Solution  Do the expressions have a common denominator? No. Rewrite each expression with the LCD. Find the LCD.  Rewrite each rational expression as an equivalent rational expression with the LCD.  Simplify the numerators.  Add the rational expressions.  Simplify the numerator.  Factor the numerator.  There are no factors common to the numerator and denominator. The fraction cannot be simplified.

Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as $$\cfrac{2a\left(2a-b\right)}{b\left(2a+b\right)\left(2a-b\right)}$$ to be able to add it to $$\cfrac{3a·b}{\left(2a+b\right)\left(2a-b\right)·b}$$. Simplify only after you have combined the numerators.

Example

Add: $$\cfrac{8}{{x}^{2}-2x-3}+\cfrac{3x}{{x}^{2}+4x+3}.$$

Solution  Do the expressions have a common denominator? No. Rewrite each expression with the LCD. Find the LCD.  Rewrite each rational expression as an equivalent fraction with the LCD.  Simplify the numerators.  Add the rational expressions.  Simplify the numerator.  The numerator is prime, so there are no common factors.