Adding and Subtracting Rational Expressions

In adding and subtracting rational expressions, it is important to remember to keep the concept and process very simple in your mind. So, remember that just like with "simple" fractions it is necessary to have common denominators in both rational expressions before you can add or subtract.

When there are common denominators, remember that you add and/or subtract the numerators only; combining like terms in the two numerators. When you subtract the second numerator, it is a distributive property problem, and then a combining like terms problem.

The following video will introduce, discuss, and model the concepts involved in adding and subtracting rational expressions.

Now, what happens when we are trying and/or subtract rational expressions without "common" or "like" denominators? Consider the problem equation image indicator $LaTeX: \frac{3}{10}+\frac{1}{6}$ . What is the least common denominator (LCD)? Remember in finding the LCD, you are really finding the "least common multiple" (LCM) of both denominators. In this problem, the LCD equals 30. Once we find the LCD, then we need to change each individual fraction to be equivalent to respective fractions that have a denominator equal to 30. Once you have common denominators in all fractions, then you add the numerators together and simplify if possible.

equation image indicator $LaTeX: \frac{3}{10}\cdot\frac{?}{?}=\frac{?}{30}\longrightarrow\frac{3}{10}\cdot\frac{3}{3}=\frac{9}{30} \\ \frac{1}{6}\cdot\frac{?}{?}=\frac{?}{30}\longrightarrow\frac{1}{6}\cdot\frac{5}{5}=\frac{5}{30} \\ \frac{9}{30}+ \frac{5}{30}=\frac{14}{30}\div \frac{2}{2}=\frac{7}{15} \\$

When thinking about denominators like x + 2 or x - 3 it becomes important to understand what makes a LCD. In the problem above, we found the LCD was equal to 30 because it is the smallest number that both 10 and 6 divide into, but how do we create that number if it isn't obvious? (Hint: Think about prime factorization.)

When dealing with rational expressions, factoring is key. We must find all of the factors of each denominator to know what the LCD should be.

Let's try one example.

Find the LCD for equation image indicator $LaTeX: \frac{3}{5a}\:\&\:\frac{b}{4a^2}$ .

equation image indicator $LaTeX: 5a=5\cdot a \\ 4a^2=4\cdot a\cdot a$

The only common factor is a, so then we would multiply all "leftover" factors: equation image indicator $LaTeX: a\cdot5\cdot4\cdot a=20a^2$ .

So the LCD would be equation image indicator LaTeX: 20a^2 .

If we are going to add those two expressions, having the LCD is extremely important.

equation image indicator $LaTeX: \frac{3}{5a^2}+\frac{b}{4a^2}=-\frac{12}{20a^2}+\frac{5b}{20a^2}=\frac{12a+5b}{20a^2}$ .

Because it is so important to find the LCD in adding and subtracting rational expressions, watch the following videos that review and apply the concept of finding the LCD (LCM in denominators).

Lowest Common Denominator Practice

Now, meticulously watch and take notes on the following videos that discuss and model the concepts involved in adding and subtracting rational expressions with monomials, binomials, multiple variables, and other unknown expressions. After watching these teaching videos, practice items are provided for you to complete to master these concepts.

Adding Rational Expressions Practice

Subtracting Rational Expressions Practice

Complex Fractions

There are times that you come across problems that involve "nested fractions" (fractions within fractions). In trying to simplify these types of "complex fractions," often times you will need to use all of the operational properties of rational expressions that you have learned to this point.

Watch the following video describing the process of simplifying complex fractions, and after the video there are some practice problems provided for you to work to attain mastery.

Complex Fractions and Nested Rational Expressions Practice

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