Unit 11.4 Complex Fractions
Rational Complex Fractions
A complex fraction is a fraction in which the numerator, the denominator, or both are themselves fractions. Simplifying complex fractions involves converting them into simple fractions. For rational complex fractions, the numerator and denominators are both rational expressions. In simplest terms, it is another way to write division of rational expressions
[latex]\frac{\left(\frac{a+b}{c+d}\right)}{\left(\frac{w+x}{y+z}\right)}=\ \frac{a+b}{c+d}\ \div\ \frac{w+x}{y+z}[/latex]
There are a few different methods to simplifying a complex fraction.
Simplify the Complex Fraction [latex]\frac{\left(\frac{1}{x+3}-\frac{1}{x}\right)}{\left(1+\frac{3}{x}\right)}[/latex]
Simplifying a complex fraction: Method 1
Rewrite the complex fraction as a division of rational expressions
[latex]\frac{\left(\frac{1}{x+3}-\frac{1}{x}\right)}{\left(1+\frac{3}{x}\right)}[/latex]
[latex]\frac{\left(\frac{1}{x+3}-\frac{1}{x}\right)}{\left(1+\frac{3}{x}\right)}\ =\left(\frac{1}{x+3}-\frac{1}{x}\right)\ \div\ \left(1+\frac{3}{x}\right)[/latex]
Simplify the numerator and denominators first
Numerator
[latex]\frac{1}{x+3}\left(\frac{x}{x}\right)-\frac{1}{x}\left(\frac{x+3}{x+3}\right)[/latex]
[latex]\left(\frac{x}{x(x+3)}-\frac{x+3}{x(x+3)}\right)[/latex]
[latex]\frac{x\ -\ x\ -\ 3}{x(x+3)}[/latex]
[latex]\frac{-3}{x(x+3)}[/latex]
Denominator
[latex]\left(1\ \left(\frac{x}{x}\right)+\frac{3}{x}\right)[/latex]
[latex]\left(\frac{x}{x}+\frac{3}{x}\right)[/latex]
[latex]\left(\frac{x+3}{x}\right)[/latex]
Rewrite
[latex]\frac{-3}{x(x+3)}[/latex] ÷ [latex]\frac{x+3}{x}[/latex]
Find the reciprocal of the second fraction and multiply
[latex]\frac{-3}{x(x+3)}\cdot\frac{x}{x + 3}[/latex]
[latex]\frac{-3x}{x(x+3)(x+3)}[/latex]
Cancel out like terms and simplify
[latex]\frac{-3}{(x+3)(x+3)}[/latex]
[latex]\frac{-3}{{(x+3)}^2}[/latex]
Simplifying a complex fraction: Method 2
Multiply the numerator and denominator by a LCM of the numerator
[latex]\frac{\frac{1}{x+3}-\frac{1}{x}}{1+\frac{3}{x}}[/latex]
The LCM of the numerator is x(x + 3)
[latex]\frac{\left(\frac{1}{x+3}-\frac{1}{x}\right)\cdot x\left(x+3\right)}{\left(1+\frac{3}{x}\right)\cdot x\left(x+3\right)}[/latex]
Multiply and simplify by canceling out like terms
[latex]\frac{\frac{1}{\cancel{x+3}}\cdot\frac{x\left(x+3\right)}{1}-\frac{1}{x}\cdot\frac{x\left(x+3\right)}{1}}{1\cdot x\left(x+3\right)+\frac{3}{x}\cdot x\left(x+3\right)}[/latex]
[latex]\frac{x-\left(x+3\right)}{x\left(x+3\right)+3\left(x+3\right)}[/latex]
Simplify
[latex]\frac{x-x-3}{\left(x+3\right)\left(x+3\right)}[/latex]
[latex]\frac{-3}{\left(x+3\right)^2}[/latex]