Rewrite the complex fraction as a division problem.

[latex]\displaystyle\large \frac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}=\normalsize\frac{12}{35}\div \frac{6}{7}[/latex]

Rewrite the division as multiplication and take the reciprocal of the divisor.

[latex]=\frac{12}{35}\cdot \frac{7}{6}[/latex]

Factor the numerator and denominator looking for common factors before multiplying numbers together.

[latex]\begin{array}{l}=\frac{2\cdot 6\cdot 7}{5\cdot 7\cdot 6}\\\\=\frac{2}{5}\cdot \frac{6\cdot 7}{6\cdot 7}\\\\=\frac{2}{5}\cdot 1\end{array}[/latex]

[latex]\displaystyle\Large \frac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}=\normalsize\frac{2}{5}[/latex]

First combine the numerator and denominator by adding or subtracting. You may need to find a common denominator first. Note that we do not show the steps for finding a common denominator, so please review that in the previous section if you are confused.

[latex]\displaystyle\Large \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}=\frac{\,\frac{5}{4}\,}{\,\frac{7}{10}\,}[/latex]

Rewrite the complex fraction as a division problem.

[latex]\displaystyle\Large \frac{\,\,\frac{5}{4}\,\,}{\,\,\frac{7}{10}\,\,}=\normalsize\frac{5}{4}\div \frac{7}{10}[/latex]

Rewrite the division as multiplication and take the reciprocal of the divisor.

[latex]=\dfrac{5}{4}\cdot \frac{10}{7}[/latex]

Multiply and simplify as needed.

[latex]\dfrac{5}{4}\cdot \frac{10}{7}=\frac{5\cdot5\cdot2}{2\cdot2\cdot7}=\frac{25}{14}[/latex]

[latex]\displaystyle\Large \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}=\normalsize\frac{25}{14}[/latex]

Begin by combining the expressions in the numerator into one expression.

Now the numerator is a single rational expression and the denominator is a single rational expression.

We can rewrite this as division and then multiplication.

State the quotient in simplest form. Rewrite division as multiplication by the reciprocal.

[latex]\frac{5x^{2}}{9}\cdot\frac{27}{15x^{3}}[/latex]

Factor the numerators and denominators.

[latex]\frac{5\cdot{x}\cdot{x}}{3\cdot3}\cdot\frac{3\cdot3\cdot3}{5\cdot3\cdot{x}\cdot{x}\cdot{x}}[/latex]

Cancel common factors.

[latex]\large\begin{array}{c}\frac{\cancel{5}\cdot{\cancel{x}}\cdot{\cancel{x}}}{\cancel{3}\cdot\cancel{3}}\cdot\frac{\cancel{3}\cdot\cancel{3}\cdot\cancel{3}}{\cancel{5}\cdot\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}\cdot{x}}=\frac{1}{x}\end{array}[/latex]

[latex]\displaystyle \frac{5{{x}^{2}}}{9}\div \frac{15{{x}^{3}}}{27}=\frac{1}{x}[/latex]

Rewrite the complex rational expression as a division problem.

[latex]=\frac{x+5}{{{x}^{2}}-16}\div \frac{{{x}^{2}}-25}{x-4}[/latex]

Rewrite the division as multiplication and take the reciprocal of the divisor. Note that the excluded values for this are [latex]-4[/latex], [latex]4[/latex], [latex]-5[/latex], and [latex]5[/latex], because those values make the denominators of one of the fractions zero.

[latex]=\frac{x+5}{{{x}^{2}}-16}\cdot \frac{x-4}{{{x}^{2}}-25}[/latex]

Factor the numerator and denominator, looking for common factors. In this case, [latex]x+5[/latex] and [latex]x–4[/latex] are common factors of the numerator and denominator.

[latex]\begin{array}{l}=\frac{(x+5)(x-4)}{(x+4)(x-4)(x+5)(x-5)}\\\\=\frac{1}{(x+4)(x-5)}\end{array}[/latex]

[latex]\displaystyle\Large \frac{\,\,\frac{x+5}{{{x}^{2}}-16}\,}{\,\,\frac{{{x}^{2}}-25}{x-4}\,}\normalsize=\frac{1}{(x+4)(x-5)},x\ne -4,4,-5,5[/latex]

Combine the expressions in the numerator and denominator. To do this, rewrite the expressions using a common denominator. There is an excluded value of [latex]0[/latex] because this makes the denominators of the fractions zero.

[latex]\begin{array}{l}=\frac{\frac{{{x}^{2}}}{{{x}^{2}}}-\frac{9}{{{x}^{2}}}}{\frac{{{x}^{2}}}{{{x}^{2}}}+\frac{5x}{{{x}^{2}}}+\frac{6}{{{x}^{2}}}}\\\\=\frac{\frac{{{x}^{2}}-9}{{{x}^{2}}}}{\frac{{{x}^{2}}+5x+6}{{{x}^{2}}}}\end{array}[/latex]

Rewrite the complex rational expression as a division problem. (When you are comfortable with the step of rewriting the complex rational fraction as a division problem, you might skip this step and go straight to rewriting it as multiplication.)

[latex]=\frac{{{x}^{2}}-9}{{{x}^{2}}}\div \frac{{{x}^{2}}+5x+6}{{{x}^{2}}}[/latex]

Rewrite the division as multiplication and take the reciprocal of the divisor.

[latex]=\frac{{{x}^{2}}-9}{{{x}^{2}}}\cdot \frac{{{x}^{2}}}{{{x}^{2}}+5x+6}[/latex]

Factor the numerator and denominator looking for common factors. In this case, [latex]x+3[/latex] and [latex]x^{2}[/latex] are common factors. We can now see there are two additional excluded values, [latex]-2[/latex] and [latex]-3[/latex].

[latex]\begin{array}{l}=\frac{(x+3)(x-3){{x}^{2}}}{{{x}^{2}}(x+3)(x+2)}\\\\=\frac{(x-3)}{(x+2)}\cdot \frac{{{x}^{2}}(x+3)}{{{x}^{2}}(x+3)}\end{array}[/latex]

[latex]\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}=\frac{x-3}{x+2},x\ne -3,-2,0[/latex]

Before combining the expressions, find a common denominator for all of the rational expressions. In this case, [latex]x^{2}[/latex] is a common denominator. Multiply by [latex]1[/latex] in the form of a fraction with the common denominator in both the numerator and denominator. In this case, multiply by [latex]\frac{{{x}^{2}}}{{{x}^{2}}}[/latex]. There is an excluded value of [latex]0[/latex] because this makes the denominators of the fractions zero.

[latex]\begin{array}{l}=\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}\cdot \frac{{{x}^{2}}}{{{x}^{2}}}\\\\=\frac{\left( 1-\frac{9}{{{x}^{2}}} \right){{x}^{2}}}{\left( 1+\frac{5}{x}+\frac{6}{{{x}^{2}}} \right){{x}^{2}}}\\\\=\frac{{{x}^{2}}-9}{{{x}^{2}}+5x+6}\end{array}[/latex]

Notice that the expression is no longer complex! You can simplify by factoring and identifying common factors. We can now see there are two additional excluded values, [latex]-2[/latex] and [latex]-3[/latex].

[latex]\begin{array}{l}=\frac{(x+3)(x-3)}{(x+3)(x+2)}\\\\=\frac{x+3}{x+3}\cdot \frac{x-3}{x+2}\\\\=1\cdot \frac{x-3}{x+2}\end{array}[/latex]

[latex]\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}=\frac{x-3}{x+2},x\ne -3,-2,0[/latex]

First, find the LCD of the denominators that occur in the numerator of the complex fraction.

If necessary, factor the denominators before finding the LCD. In this problem, there is just one denominator, [latex]y[/latex]. The LCD of the numerator is [latex]y[/latex].

Multiply the [latex]y[/latex] in the numerator by [latex]\frac{y}{y}[/latex], so that the denominators match the LCD.

[latex]\dfrac{\dfrac{7}{y}+y}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7}{y}+y \cdot \dfrac{y}{y}}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7}{y}+\dfrac{y^2}{y}}{\dfrac{5}{y}-y}[/latex]

Add the fractions in the numerator.

[latex]\dfrac{\dfrac{7}{y}+\dfrac{y^2}{y}}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-y}[/latex]

At this point, repeat the preceding steps for the denominator of the complex fraction.

[latex]\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-y}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-y \cdot \dfrac{y}{y}}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5}{y}-\dfrac{y^2}{y}}=\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5-y^2}{y}}[/latex]

You are now ready to divide the numerator by the denominator. To accomplish this, multiply the numerator by the reciprocal of the denominator.

[latex]\dfrac{\dfrac{7+y^2}{y}}{\dfrac{5-y^2}{y}}=\dfrac{7+y^2}{y} \cdot \dfrac{y}{5-y^2}[/latex]

The [latex]y[/latex] on the top and bottom cancel, therefore:
[latex]\dfrac{\dfrac{7}{y}+y}{\dfrac{5}{y}-y}=\dfrac{7+y^2}{5-y^2}[/latex]