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As we know, a rational number is one that can be expressed as a fraction. That is to say,

$\frac{a}{b}$

where a and b are integers and $b\ne 0$ .

A rational expression, which is also called an algebraic fraction, can be expressed as a quotient of polynomials. That is to say,

$\frac{a}{b}$

where a and b are polynomials and $b\ne 0$ .

We can add and subtract rational expressions easily, but there are a few steps that we must take when the expressions have unlike denominators. First, let's look at adding a rational expression with like denominators.

**Example 1**

Add the following rational expressions:

$\frac{4}{15ab}+\frac{7}{15ab}$

To add rational expressions with like denominators, simply add the numerators.

So $\frac{4}{15ab}+\frac{7}{15ab}=\frac{11}{15ab}$

There are a few steps you must take with rational expressions that have unlike denominators when you add or subtract them. They are as follows:

- The first step is to find the LCM, or least common multiple, of the denominator. Another name for the LCM of a fraction or rational expression is the least common denominator, or LCD.
- Write each expression using the LCD. Make sure that each term has the LCD as the denominator.
- Add or subtract the numerators as the equation requires.
- Simplify the answer if necessary.

**Example 2**

Add $\left(\frac{1}{4a}\right)+\left(\frac{1}{5b}\right)$ .

Since the denominators are not the same, we must first find the least common denominator.

Since 4a and 5b have no common factors, the LCM is simply their product: $4a\times 5b$ .

That is to say, the LCD of the fractions is 20ab.

Rewrite the fractions using the LCD.

$\left(\frac{1}{4a}\right)\times \left(\frac{5b}{5b}\right)+(\frac{1}{\mathrm{5b}}\times (\frac{\mathrm{4a}}{\mathrm{4a}})$

$=\frac{\mathrm{5b}}{20ab}+\frac{\mathrm{4a}}{20ab}$

$=\frac{4a+5b}{20ab}$

**Example 3**

Add $\left(\frac{1}{6{x}^{2}}\right)+\left(\frac{5}{8x{y}^{2}}\right)$ .

Since the denominators are not the same, first we must find the LCD.

Here, the greatest common factor (GCF) of $6{x}^{2}$ and $8x{y}^{2}$ is $2x$ . So to find the LCM, we must divide the product by $2x$ .

$\mathrm{LCM}=\frac{\left(6{x}^{2}\right)\left(8x{y}^{2}\right)}{2x}$

$=\frac{(2\times 3\times x\times x\times 8x{y}^{2})}{2x}$

$=3\times x\times {8xy}^{2}$

$=24{x}^{2}{y}^{2}$

Next, we rewrite the fractions using the LCD.

$(\frac{1}{6{x}^{2}}\times \frac{4x{y}^{2}}{4x{y}^{2}})+(\frac{5}{8x{y}^{2}}\times \frac{3}{x})$

Finally, we simplify by performing the calculations.

$=\frac{4x{y}^{2}}{24{x}^{2}{y}^{2}}+\frac{15}{24{x}^{2}{y}^{2}}$

$=\frac{4x{y}^{2}+15x}{24{x}^{2}{y}^{2}}$

**Example 4**

Subtract $\left(\frac{4}{a}\right)\u2013\left(\frac{6}{a\u20135}\right)$

Since the denominators are not the same, we must find the LCD.

The LCM of $a$ and $a-5$ is $a(a\u20135)$ .

That is to say, the LCD of $a$ and $a-5$ is $a(a\u20135)$ .

So we will rewrite the fractions using the LCD.

$\left(\frac{4}{a}\right)\u2013\left(\frac{6}{a\u20135}\right)=\frac{4\left(a\u20135\right)}{a(a\u20135)}\u2013\frac{6a}{a(a\u20135)}$

Simplify the numerator on the first fraction.

$\frac{4a\u201320}{a(a\u20135)}\u2013\frac{6a}{a(a\u20135)}$

Next, we subtract the numerators.

$\frac{4a\u201320\u20136a}{a(a\u20135)}$

Then we simplify the numerator.

$\frac{-2a\u201320}{a(a\u20135)}$

Adding and Subtracting Rational Expressions with Like Denominators

Simplifying Rational Expressions

College Algebra Diagnostic Tests

Tutoring is an excellent way to help your student learn the ins and outs of adding and subtracting rational expressions with unlike denominators. Let's face it-this can be tricky math for a lot of students. If your student is struggling with the steps and processes involved in adding and subtracting rational expressions with unlike denominators, a private tutor can give them the 1-on-1 assistance they need. But tutoring is not just for students who are struggling. Even for students who understand the concepts, it can be good to get extra practice or to dig deeper into the topic. We would love the opportunity to connect your student with a qualified tutor to help them understand adding and subtracting rational expressions with unlike denominators. Contact Varsity Tutors and speak with one of our friendly Educational Directors to get started today.

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