Adding and Subtracting Rational Expressions with Unlike Denominators

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 $b\ne 0$ .

Adding and subtracting rational expressions with like denominators

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

Adding and subtracting rational expressions with unlike denominators

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:

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

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

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

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

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

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

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

Next, we rewrite the fractions using the LCD.

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

Finally, we simplify by performing the calculations.

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

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

Example 4

Subtract $\left(\frac{4}{a}\right)–\left(\frac{6}{a–5}\right)$

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

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

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

So we will rewrite the fractions using the LCD.

$\left(\frac{4}{a}\right)–\left(\frac{6}{a–5}\right)=\frac{4\left(a–5\right)}{a(a–5)}–\frac{6a}{a(a–5)}$

Simplify the numerator on the first fraction.

$\frac{4a–20}{a(a–5)}–\frac{6a}{a(a–5)}$

Next, we subtract the numerators.

$\frac{4a–20–6a}{a(a–5)}$

Then we simplify the numerator.

$\frac{-2a–20}{a(a–5)}$

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