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: $${\frac{(2x - 3)(y+4) + (2x-3)(y+4)}{(2x-3)(y+4) + (x+2)(y+4)}}=$$ $$\frac{2(y+4)(2x-3)}{(y+4)(3x-1)}=$$ $$\frac{2\cancel{(y+4)}(2x-3)}{\cancel{(y+4)}(3x-1)}=$$ $$\frac{2(2x - 3)}{(3x - 1)}$$

Skills Check:

Example #1

Simplify each. $$\frac{\Large{\frac{x + 1}{x^{2}}- \frac{4}{x - 6}}}{\Large{\frac{x + 1}{x}+ \frac{x}{x - 6}}}$$

A
$$\frac{x^{3}+ x^{2}- x - 1}{4x^{3}- 32x^{2}+ 160x}$$
B
$$\frac{-x^{4}- 2x^{3}+ 63x^{2}}{28x^{2}+ 8x + 4}$$
C
$$\frac{5x^{3}- 50x + 144}{4x^{3}+ 20x^{2}- 96x}$$
D
$$\frac{-3x^{2}- 5x - 6}{2x^{3}- 5x^{2}- 6x}$$
E
$$\frac{4x^{2}}{x - 6}$$

Example #2

Simplify each. $$\frac{\Large{\frac{3x - 1}{6x + 18}+ \frac{1}{2}}}{\Large{\frac{2}{3}- \frac{3x - 1}{x + 3}}}$$

A
$$\frac{-6x + 2}{-53x + 21}$$
B
$$\frac{3x^{2}+ 8x - 147}{-27x^{2}- 70x + 33}$$
C
$$\frac{81x^{2}- 36x + 63}{21x + 53}$$
D
$$\frac{3x + 4}{-7x + 9}$$
E
$$\frac{5x - 1}{-2x + 7}$$

Example #3

Simplify each. $$\frac{\Large{\frac{3}{3x - 5}+ \frac{x + 1}{3x - 5}}}{\Large{\frac{1}{2}+ \frac{9}{x + 1}}}$$

A
$$\frac{1}{4}$$
B
$$\frac{81x - 99}{12x - 8}$$
C
$$\frac{3x - 2}{x + 4}$$
D
$$\frac{-18x + 46}{9x^{3}- 24x^{2}+ 30}$$
E
$$\frac{2x^{2}+ 10x + 8}{3x^{2}+ 52x - 95}$$