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Study Guides > College Algebra

Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression a1b+c\frac{a}{\frac{1}{b}+c} can be simplified by rewriting the numerator as the fraction a1\frac{a}{1} and combining the expressions in the denominator as 1+bcb\frac{1+bc}{b}. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get a1b1+bc\frac{a}{1}\cdot \frac{b}{1+bc}, which is equal to ab1+bc\frac{ab}{1+bc}.

How To: Given a complex rational expression, simplify it.

  1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.
  2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.
  3. Rewrite as the numerator divided by the denominator.
  4. Rewrite as multiplication.
  5. Multiply.
  6. Simplify.

Example 6: Simplifying Complex Rational Expressions

Simplify: y+1xxy\frac{y+\frac{1}{x}}{\frac{x}{y}} .

Solution

Begin by combining the expressions in the numerator into one expression.
yxx+1xMultiply by xxto get LCD as denominator.xyx+1xxy+1xAdd numerators.\begin{array}{cc}y\cdot \frac{x}{x}+\frac{1}{x}\hfill & \text{Multiply by }\frac{x}{x}\text{to get LCD as denominator}.\hfill \\ \frac{xy}{x}+\frac{1}{x}\hfill & \\ \frac{xy+1}{x}\hfill & \text{Add numerators}.\hfill \end{array}
Now the numerator is a single rational expression and the denominator is a single rational expression.
xy+1xxy\frac{\frac{xy+1}{x}}{\frac{x}{y}}
We can rewrite this as division, and then multiplication.
xy+1x÷xyxy+1xyxRewrite as multiplication.y(xy+1)x2Multiply.\begin{array}{cc}\frac{xy+1}{x}\div \frac{x}{y}\hfill & \\ \frac{xy+1}{x}\cdot \frac{y}{x}\hfill & \text{Rewrite as multiplication}\text{.}\hfill \\ \frac{y\left(xy+1\right)}{{x}^{2}}\hfill & \text{Multiply}\text{.}\hfill \end{array}

Try It 5

Simplify: xyyxy\frac{\frac{x}{y}-\frac{y}{x}}{y} Solution

Q & A

Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

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