Complex Rational Expressions Lesson

A complex fraction is a fraction that contains another fraction in its numerator, denominator, or both. $$\Large{\frac{\frac{3 + 5}{7}}{\frac{9 + 1}{11}}}$$ To simplify this type of fraction, we can use two different strategies.
Method 1:

• Simplify the numerator and denominator separately
• Perform the main division

$$\Large{\frac{\frac{3 + 5}{7}}{\frac{9 + 1}{11}}}=\Large{\frac{\frac{8}{7}}{\frac{10}{11}}}$$ $$\frac{8}{7}\cdot \frac{11}{10}=\frac{44}{35}$$ Method 2:

• Multiply the numerator and denominator of the complex fraction by the LCD of all denominators in the complex fraction
• Simplify

$$\Large{\frac{\frac{3 + 5}{7}}{\frac{9 + 1}{11}}}$$ LCD(7,11) = 77 $$\Large{\frac{\frac{3 + 5}{7}\cdot \normalsize{77}}{\frac{9 + 1}{11}\cdot \normalsize{77}}}=\normalsize{\frac{11 \cdot 3 + 11 \cdot 5}{7 \cdot 9 + 7 \cdot 1}}=\frac{88}{70}=\frac{44}{35}$$

Simplifying Complex Rational Expressions

When we come across complex rational expressions, we can use the same strategies. Method 2 is normally a bit faster, so we will use this method for our examples. To simplify a complex rational expression, we will multiply the numerator and the denominator of the complex rational expression by the LCD of all denominators involved. Once this is done, we can simplify if needed. Let's look at a few examples.
Example 1: Simplify each. $$\Large{\frac{\frac{x}{3x + 12}- \frac{x}{9}}{\frac{x}{x + 4}- \frac{x}{4}}}$$ To find our LCD, let's think about the denominators: $$3(x + 4), 9, (x+4), 4$$ LCD: $$36(x + 4)$$ Let's multiply the numerator and denominator of the complex rational expression by the LCD: $$\frac{\normalsize{36(x + 4)}\left(\Large{\frac{x}{3x + 12}- \frac{x}{9}}\right)}{36(x+4) \left(\Large{\frac{x}{x + 4}- \frac{x}{4}}\right)}$$ Numerator: $$36(x+4) \cdot \frac{x}{3(x + 4)}- 36(x + 4) \cdot \frac{x}{9}$$ $$\require{cancel}12\cancel{36(x+4)}\cdot \frac{x}{\cancel{3(x + 4)}}- 4\cancel{36}(x + 4) \cdot \frac{x}{\cancel{9}}$$ $$12x - 4x(x + 4)$$ Denominator: $$36(x + 4) \cdot \frac{x}{x + 4}- 36(x + 4) \cdot \frac{x}{4}$$ $$36\cancel{(x + 4)}\cdot \frac{x}{\cancel{(x + 4)}}- 9\cancel{36}(x + 4) \cdot \frac{x}{\cancel{4}}$$ $$36x - 9x(x + 4)$$ Our complex rational expression becomes: $$\frac{12x - 4x(x + 4)}{36x - 9x(x + 4)}$$ $$\frac{-4x^2 - 4x}{-9x^2}$$ $$\frac{\cancel{-x}(4x + 4)}{\cancel{-x}\cdot 9x}$$ $$\frac{4x + 4}{9x}$$ Example 2: Simplify each. $$\Large{\frac{\frac{4}{x + 2}+ \frac{x + 2}{x - 1}}{\frac{4}{x - 1}- \frac{x - 1}{2}}}$$ To find our LCD, let's think about the denominators: $$2,(x + 2), (x - 1)$$ LCD: $$2(x + 2)(x - 1)$$ Let's multiply the numerator and denominator of the complex rational expression by the LCD: $$\frac{2(x + 2)(x - 1) \left(\Large{\frac{4}{x + 2}+ \frac{x + 2}{x - 1}}\right)}{2(x + 2)(x - 1)\left(\Large{\frac{4}{x - 1}- \frac{x - 1}{2}}\right)}$$ Numerator: $$2(x + 2)(x - 1) \cdot \frac{4}{(x + 2)}+ 2(x + 2)(x - 1) \cdot \frac{(x + 2)}{(x - 1)}$$ $$2\cancel{(x + 2)}(x - 1) \cdot \frac{4}{\cancel{(x + 2)}}+ 2(x + 2)\cancel{(x - 1)}\cdot \frac{(x + 2)}{\cancel{(x - 1)}}$$ $$8(x - 1) + 2(x + 2)(x + 2)$$ Denominator: $$2(x + 2)(x - 1) \cdot \frac{4}{(x - 1)}- 2(x + 2)(x - 1) \cdot \frac{(x - 1)}{2}$$ $$2(x + 2)\cancel{(x - 1)}\cdot \frac{4}{\cancel{(x - 1)}}- \cancel{2}(x + 2)(x - 1) \cdot \frac{(x - 1)}{\cancel{2}}$$ $$8(x + 2) - (x + 2)(x - 1)(x - 1)$$ Our complex rational expression becomes: $$\frac{8(x - 1) + 2(x + 2)(x + 2)}{8(x + 2) - (x + 2)(x - 1)(x - 1)}$$ $$\frac{8x - 8 + 2x^2 + 8x + 8}{8x + 16 - x^3 + 3x - 2}$$ $$\frac{2x^2 + 16x}{-x^3 + 11x + 14}$$ $$\frac{2x(x + 8)}{-(x + 2)(x^2 - 2x - 7)}$$

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