Lesson Objectives
• Demonstrate an understanding of how to simplify a complex fraction
• Demonstrate an understanding of how to find the LCD for a group of rational expressions
• Learn how to simplify a complex rational expression using the LCD Method

## How to Simplify a Complex Rational Expression

In our pre-algebra course, we learned the definition of a complex fraction. A complex fraction contains a fraction in its numerator, denominator or both. Let's take a look at an example: $$\Large{\frac{\frac{1}{3} + \frac{5}{12}}{\frac{9}{20} + \frac{2}{9}}}$$ We can simplify this complex fraction by multiplying the numerator and denominator by the LCD of all fractions. Our complex fraction contains the fractions: $$\frac{1}{3}, \frac{5}{12}, \frac{9}{20}, \frac{2}{9}$$ The LCD is the LCM of the denominators:
LCM(3,9,12,20) = 180
We can multiply the numerator and denominator of the complex fraction by 180. Essentially, we are multiplying each fraction by 180: $$\frac{180}{180} \cdot \Large{\frac{\frac{1}{3} + \frac{5}{12}}{\frac{9}{20} + \frac{2}{9}}}$$ If we distribute 180 to each fraction: $$\require{cancel}180 \cdot \frac{1}{3} = \frac{60\cancel{180}}{\cancel{3}} = 60$$ $$180 \cdot \frac{5}{12} = \frac{15\cancel{180} \cdot 5}{\cancel{12}} = 75$$ $$180 \cdot \frac{9}{20} = \frac{9\cancel{180} \cdot 9}{\cancel{20}} = 81$$ $$180 \cdot \frac{2}{9} = \frac{20\cancel{180} \cdot 2}{\cancel{20}} = 40$$ Now we can finish simplifying our complex fraction: $$\frac{60 + 75}{81 + 40} = \frac{135}{121}$$

### Simplifying a Complex Rational Expression

When we simplify a complex rational expression, we multiply the numerator and denominator by the LCD of all rational expressions. Let's look at a few examples.
Example 1: Simplify each $$\frac{\Large{\frac{x \hspace{.2em}+\hspace{.2em} 6}{25} \hspace{.2em}+\hspace{.2em} \frac{x \hspace{.2em}+ \hspace{.2em}6}{x \hspace{.2em}-\hspace{.2em} 4}}}{x + 6}$$ Step 1) Find the LCD of all rational expressions:
LCD » 25(x - 4)
Step 2) Multiply the LCD by the numerator and denominator of the complex rational expression. Essentially, we are multiplying each rational expression by 25(x - 4): $$25(x-4) \cdot \frac{x + 6}{25} = \cancel{25}(x-4) \cdot \frac{x + 6}{\cancel{25}} = (x-4)(x + 6)$$ $$25(x-4) \cdot \frac{x + 6}{25} =$$$$\cancel{25}(x-4) \cdot \frac{x + 6}{\cancel{25}} =$$$$(x-4)(x + 6)$$ $$25(x-4) \cdot \frac{x + 6}{x - 4} = 25\cancel{(x-4)} \cdot \frac{x + 6}{\cancel{(x - 4)}} = 25(x + 6)$$ $$25(x-4) \cdot \frac{x + 6}{x - 4} =$$$$25\cancel{(x-4)} \cdot \frac{x + 6}{\cancel{(x - 4)}} =$$$$25(x + 6)$$ $$25(x-4) \cdot (x + 6) = 25(x - 4)(x + 6)$$ $$25(x-4) \cdot (x + 6) =$$$$25(x - 4)(x + 6)$$ Step 3) We will simplify our complex rational expression: $$\frac{(x-4)(x + 6) + 25(x + 6)}{25(x - 4)(x + 6)}$$ $$\frac{(x + 6)(x + 21)}{25(x - 4)(x + 6)} =$$ $$\frac{\cancel{(x + 6)}(x + 21)}{25(x - 4)\cancel{(x + 6)}} =$$ $$\frac{(x + 21)}{25(x - 4)}$$ Example 2: Simplify each $$\Large{\frac{\frac{y \hspace{.2em}+\hspace{.2em} 4}{x \hspace{.2em}+\hspace{.2em} 2} + \frac{y \hspace{.2em}+\hspace{.2em} 4}{x \hspace{.2em}+\hspace{.2em} 2}}{\frac{y \hspace{.2em}+\hspace{.2em} 4}{x \hspace{.2em}+\hspace{.2em} 2} + \frac{y \hspace{.2em}+ \hspace{.2em}4}{2x \hspace{.2em}-\hspace{.2em} 3}}}$$ Step 1) Find the LCD of all rational expressions: LCD » (x + 2)(2x - 3)
Step 2) Multiply the LCD by the numerator and denominator of the complex rational expression. Essentially, we are multiplying each rational expression by (x + 2)(2x - 3): $$(x + 2)(2x - 3) \cdot \frac{y + 4}{x + 2} = \cancel{(x + 2)}(2x - 3) \cdot \frac{y + 4}{\cancel{(x + 2)}} = (2x - 3)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{x + 2} =$$$$\cancel{(x + 2)}(2x - 3) \cdot \frac{y + 4}{\cancel{(x + 2)}} =$$$$(2x - 3)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{x + 2} = \cancel{(x + 2)}(2x - 3) \cdot \frac{y + 4}{\cancel{(x + 2)}} = (2x - 3)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{x + 2} =$$$$\cancel{(x + 2)}(2x - 3) \cdot \frac{y + 4}{\cancel{(x + 2)}} =$$$$(2x - 3)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{x + 2} = \cancel{(x + 2)}(2x - 3) \cdot \frac{y + 4}{\cancel{(x + 2)}} = (2x - 3)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{x + 2} =$$$$\cancel{(x + 2)}(2x - 3) \cdot \frac{y + 4}{\cancel{(x + 2)}} =$$$$(2x - 3)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{2x - 3} = (x + 2)\cancel{(2x - 3)} \cdot \frac{y + 4}{\cancel{(2x - 3)}} = (x + 2)(y + 4)$$ $$(x + 2)(2x - 3) \cdot \frac{y + 4}{2x - 3} =$$$$(x + 2)\cancel{(2x - 3)} \cdot \frac{y + 4}{\cancel{(2x - 3)}} =$$$$(x + 2)(y + 4)$$ Step 3) We will simplify our complex rational expression: $${\frac{(2x - 3)(y+4) + (2x-3)(y+4)}{(2x-3)(y+4) + (x+2)(y+4)}} =$$ $$\frac{2(y+4)(2x-3)}{(y+4)(3x-1)}=$$ $$\frac{2\cancel{(y+4)}(2x-3)}{\cancel{(y+4)}(3x-1)} =$$ $$\frac{2(2x - 3)}{(3x - 1)}$$