8.5 Simplify complex rational expressions  (Page 2/3)

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

$\frac{\frac{3b}{b-5}}{\frac{b^2}{b^2-25}}$

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

$\frac{\frac{8}{d^2+9d+18}}{\frac{12}{d+6}}$

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

$\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}$

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

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

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

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

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

$\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{v^2}-\frac{1}{w^2}}$

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

$\frac{y-\frac{2y}{y-4}}{\frac{2}{y-4}-\frac{2}{y+4}}$

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

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

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

$\frac{\frac{1}{4}+\frac{1}{9}}{\frac{1}{6}+\frac{1}{12}}$

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

$\frac{\frac{1}{6}+\frac{4}{15}}{\frac{3}{5}-\frac{1}{2}}$

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

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

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

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

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

$\frac{\frac{5}{y-4}}{\frac{3}{y+4}+\frac{2}{y^2-16}}$

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

$\frac{\frac{3}{s+6}+\frac{5}{s-6}}{\frac{1}{s^2-36}+\frac{4}{s+6}}$

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

$\frac{\frac{5}{b^2-6b-27}}{\frac{3}{b-9}+\frac{1}{b+3}}$

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

$\frac{\frac{6}{d-4}-\frac{2}{d+7}}{\frac{2d}{d^2+3d-28}}$

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

$\frac{\frac{n}{n-2}}{3+\frac{5}{n-2}}$

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

$\frac{7+\frac{2}{q-2}}{\frac{1}{q+2}}$

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

$\frac{\frac{v}{w}+\frac{1}{v}}{\frac{1}{v}-\frac{v}{w}}$

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

$\frac{\frac{3}{b^2-3b-40}}{\frac{5}{b+5}-\frac{2}{b-8}}$

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

$\frac{\frac{2}{r-9}}{\frac{1}{r+9}+\frac{3}{r^2-81}}$

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

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

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 $R_2=12$ .

Ironing Lenore can do the ironing for her family’s business in $h$ hours. Her daughter would take $h+2$ hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is $\frac{1}{\frac{1}{h}+\frac{1}{h+2}}$ .

1. ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{h}+\frac{1}{h+2}}$ .
2. ⓑ Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if $h=4$ .

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?

Efraim wants to start simplifying the complex fraction $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{a}-\frac{1}{b}}$ by cancelling the variables from the numerator and denominator. Explain what is wrong with Efraim’s plan.