# Mixed Expressions

A mixed expression, commonly called a complex fraction , is a fraction where the numerator, the denominator or both the numerator and denominator contain a fraction.

We can use the rules of algebra to manipulate these expressions into rational ones.

Example 1:

Simplify:

$\frac{\left(\frac{3x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}\right)}{\left(2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{x}{4x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2}\right)}$

First, rewrite the denominator as a single rational expression.

$\begin{array}{l}\frac{\left(\frac{3x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}\right)}{\left(2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{x}{4x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2}\right)}=\frac{\left(\frac{3x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}\right)}{\left(\frac{\left(8x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4\right)+x}{4x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\left(\frac{3x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}\right)}{\left(\frac{9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4}{4x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2}\right)}\end{array}$

Since the fraction line means divide, rewrite the problem as multiplication by the reciprocal .

$\begin{array}{l}=\frac{3x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}\cdot \frac{4x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2}{9x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4}\\ =\frac{12{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}6x}{9{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}5x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4}\end{array}$

This is a rational expression.

A second method we can use is to multiply the numerator and denominator of the original improper fraction by the LCD of the two fractions (which will be another name for $1$ ).

Example 2:

Simplify:

$\frac{\left(\frac{3x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}\right)}{\left(\frac{5}{2x}\right)}$

Multiply both numerator and denominator by $2x\left(x+1\right)$ .

$\begin{array}{l}=\frac{\left(\frac{3x}{\overline{)x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}2x\overline{)\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1\right)}\right)}{\left(\frac{5}{\overline{)2x}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\overline{)2x}\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1\right)\right)}\\ =\frac{3x\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}2x}{5\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1\right)}\\ =\frac{6{x}^{2}}{5x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}5}\end{array}$