8.5 Simplify Complex Rational Expressions

If you miss a problem, go back to the section listed and review the material.

Simplify: $\frac{\frac{3}{5}}{\frac{9}{10}} .$
If you missed this problem, review the reference.

Simplify: $\frac{1 - \frac{1}{3}}{4^{2} + 4 \cdot 5} .$
If you missed this problem, review the reference.

Simplify: $\frac{\frac{4}{y - 3}}{\frac{8}{y^{2} - 9}} .$

Simplify: $\frac{\frac{2}{x^{2} - 1}}{\frac{3}{x + 1}} .$

Simplify: $\frac{\frac{1}{x^{2} - 7 x + 12}}{\frac{2}{x - 4}} .$

Simplify: $\frac{\frac{1}{2} + \frac{2}{3}}{\frac{5}{6} + \frac{1}{12}} .$

Simplify: $\frac{\frac{3}{4} - \frac{1}{3}}{\frac{1}{8} + \frac{5}{6}} .$

Simplify: $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}} .$

Simplify: $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^{2}} - \frac{1}{b^{2}}} .$

Simplify: $\frac{b - \frac{3 b}{b + 5}}{\frac{2}{b + 5} + \frac{1}{b - 5}} .$

Simplify: $\frac{1 - \frac{3}{c + 4}}{\frac{1}{c + 4} + \frac{c}{3}} .$

Simplify: $\frac{\frac{1}{2} + \frac{1}{5}}{\frac{1}{10} + \frac{1}{5}} .$

Simplify: $\frac{\frac{1}{4} + \frac{3}{8}}{\frac{1}{2} - \frac{5}{16}} .$

Simplify: $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{a}{b} + \frac{b}{a}} .$

Simplify: $\frac{\frac{1}{x^{2}} - \frac{1}{y^{2}}}{\frac{1}{x} + \frac{1}{y}} .$

Simplify: $\frac{\frac{3}{x + 2}}{\frac{5}{x - 2} - \frac{3}{x^{2} - 4}} .$

Simplify: $\frac{\frac{2}{x - 7} - \frac{1}{x + 7}}{\frac{6}{x + 7} - \frac{1}{x^{2} - 49}} .$

Simplify: $\frac{\frac{3}{x^{2} + 7 x + 10}}{\frac{4}{x + 2} + \frac{1}{x + 5}} .$

Simplify: $\frac{\frac{4}{y + 5} + \frac{2}{y + 6}}{\frac{3 y}{y^{2} + 11 y + 30}} .$

Simplify: $\frac{\frac{x}{x + 3}}{1 + \frac{1}{x + 3}} .$

Simplify: $\frac{1 + \frac{1}{x - 1}}{\frac{3}{x + 1}} .$

Simplify.

$\frac{\frac{2 a}{a + 4}}{\frac{4 a^{2}}{a^{2} - 16}}$

Simplify.

$\frac{\frac{5}{c^{2} + 5 c - 14}}{\frac{10}{c + 7}}$

Simplify.

$\frac{\frac{1}{2} + \frac{5}{6}}{\frac{2}{3} + \frac{7}{9}}$

Simplify.

$\frac{\frac{2}{3} - \frac{1}{9}}{\frac{3}{4} + \frac{5}{6}}$

Simplify.

$\frac{\frac{n}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m}}$

Simplify.

$\frac{\frac{1}{r} + \frac{1}{t}}{\frac{1}{r^{2}} - \frac{1}{t^{2}}}$

Simplify.

$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Simplify.

$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$

Simplify.

$\frac{\frac{1}{3} + \frac{1}{8}}{\frac{1}{4} + \frac{1}{12}}$

Simplify.

$\frac{\frac{5}{6} + \frac{2}{9}}{\frac{7}{18} - \frac{1}{3}}$

Simplify.

$\frac{\frac{c}{d} + \frac{1}{d}}{\frac{1}{d} - \frac{d}{c}}$

Simplify.

$\frac{\frac{1}{p} + \frac{1}{q}}{\frac{1}{p^{2}} - \frac{1}{q^{2}}}$

Simplify.

$\frac{\frac{2}{x + 5}}{\frac{3}{x - 5} + \frac{1}{x^{2} - 25}}$

Simplify.

$\frac{\frac{5}{z^{2} - 64} + \frac{3}{z + 8}}{\frac{1}{z + 8} + \frac{2}{z - 8}}$

Simplify.

$\frac{\frac{4}{a^{2} - 2 a - 15}}{\frac{1}{a - 5} + \frac{2}{a + 3}}$

Simplify.

$\frac{\frac{5}{c + 2} - \frac{3}{c + 7}}{\frac{5 c}{c^{2} + 9 c + 14}}$

Simplify.

$\frac{2 + \frac{1}{p - 3}}{\frac{5}{p - 3}}$

Simplify.

$\frac{\frac{m}{m + 5}}{4 + \frac{1}{m - 5}}$

Use either method.

$\frac{\frac{3}{4} - \frac{2}{7}}{\frac{1}{2} + \frac{5}{14}}$

Use either method.

$\frac{\frac{2}{a + 4}}{\frac{1}{a^{2} - 16}}$

Use either method.

$\frac{\frac{3}{m} + \frac{3}{n}}{\frac{1}{m^{2}} - \frac{1}{n^{2}}}$

Use either method.

$\frac{x - \frac{3 x}{x + 2}}{\frac{3}{x + 2} + \frac{3}{x - 2}}$

Electronics The resistance of a circuit formed by connecting two resistors in parallel is $\frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$.

1. ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$.
2. ⓑ Find the resistance of the circuit when $R_{1} = 8$ and $R_{2} = 12$.

In this section, you learned to simplify the complex fraction $\frac{\frac{3}{x + 2}}{\frac{x}{x^{2} - 4}}$ two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?