# Adding and Subtracting Rational Expressions with Unlike Denominators

There are a few steps to follow when you add or subtract rational expressions with unlike denominators.

1. To add or subtract rational expressions with unlike denominators, first find the LCM of the denominator. The LCM of the denominators of fraction or rational expressions is also called least common denominator , or LCD.

2. Write each expression using the LCD. Make sure each term has the LCD as its denominator.

3. Add or subtract the numerators.

4. Simplify as needed.

Example 1:

Add $\frac{1}{3a}+\frac{1}{4b}$ .

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

Since $3a$ and $4b$ have no common factors, the LCM is simply their product: $3a\cdot 4b$ .

That is, the LCD of the fractions is $12ab$ .

Rewrite the fractions using the LCD.

$\begin{array}{l}\left(\frac{1}{3a}\cdot \frac{4b}{4b}\right)+\left(\frac{1}{4b}\cdot \frac{3a}{3a}\right)=\frac{4b}{12ab}+\frac{3a}{12ab}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{3a+4b}{12ab}\end{array}$

Example 2:

Add $\frac{1}{4{x}^{2}}+\frac{5}{6x{y}^{2}}$ .

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

Here, the GCF of $4{x}^{2}$ and $6x{y}^{2}$ is $2x$ . So, the LCM is the product divided by $2x$ :

$\begin{array}{l}\text{LCM}=\frac{4{x}^{2}\cdot 6x{y}^{2}}{2x}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{2\cdot \overline{)2}\cdot \overline{)x}\cdot x\cdot 6x{y}^{2}}{\overline{)2}\cdot \overline{)x}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=12{x}^{2}{y}^{2}\end{array}$

Rewrite the fractions using the LCD.

$\begin{array}{l}\frac{1}{4{x}^{2}}\cdot \frac{3x{y}^{2}}{3x{y}^{2}}+\frac{5}{6x{y}^{2}}\cdot \frac{2x}{2x}=\frac{3x{y}^{2}}{12{x}^{2}{y}^{2}}+\frac{10x}{12{x}^{2}{y}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{3x{y}^{2}+10x}{12{x}^{2}+{y}^{2}}\end{array}$

Example 3:

Subtract $\frac{2}{a}-\frac{3}{a-5}$ .

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

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

That is, the LCD of the fractions is $a\left(a-5\right)$ .

Rewrite the fraction using the LCD.

$\frac{2}{a}-\frac{3}{a-5}=\frac{2\left(a-5\right)}{a\left(a-5\right)}-\frac{3a}{a\left(a-5\right)}$

Simplify the numerator.

$=\frac{2a-10}{a\left(a-5\right)}-\frac{3a}{a\left(a-5\right)}$

Subtract the numerators.

$=\frac{2a-10-3a}{a\left(a-5\right)}$

Simplify.

$=\frac{-a-10}{a\left(a-5\right)}$

Example 4:

Add $\frac{5}{c+2}+\frac{6}{c-3}$ .

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

The LCM of $c+2$ and $c-3$ is $\left(c+2\right)\left(c-3\right)$ .

That is, the LCD of the fractions is $\left(c+2\right)\left(c-3\right)$ .

Rewrite the fraction using the LCD.

$\frac{5}{c+2}+\frac{6}{c-3}=\frac{5\left(c-3\right)}{\left(c+2\right)\left(c-3\right)}+\frac{6\left(c+2\right)}{\left(c+2\right)\left(c-3\right)}$

Simplify each numerator.

$=\frac{5c-15}{\left(c+2\right)\left(c-3\right)}+\frac{6c+12}{\left(c+2\right)\left(c-3\right)}$

$=\frac{5c-15+6c+12}{\left(c+2\right)\left(c-3\right)}$
$=\frac{11c-3}{\left(c+2\right)\left(c-3\right)}$