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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 137

Simplificar:

$\begin{matrix} \frac{a - \frac{a}{b}}{b - \frac{1}{b}} & = \frac{\frac{a b - a}{b}}{\frac{b^{2} - 1}{b}} \\ = \frac{a (b - 1)}{b^{2} - 1} \\ = \frac{a (b - 1)}{(b - 1) (b + 1)} \\ = \frac{a}{b + 1} \end{matrix}$
$\begin{matrix} \frac{x^{2} - \frac{1}{x}}{1 - \frac{1}{x}} & = \frac{\frac{x^{3} - 1}{x}}{\frac{x - 1}{x}} \\ = \frac{(x - 1) (x^{2} + x + 1)}{x - 1} \\ = x^{2} + x + 1 \end{matrix}$
$\begin{matrix} \frac{\frac{a}{b} - \frac{b}{a}}{1 + \frac{b}{a}} & = \frac{\frac{a^{2} - b^{2}}{a b}}{\frac{a + b}{a}} \\ = \frac{a^{2} - b^{2}}{b (a + b)} \\ = \frac{(a - b) (a + b)}{b (a + b)} \\ = \frac{a - b}{b} \end{matrix}$
$\begin{matrix} \frac{\frac{1}{m} + \frac{1}{n}}{\frac{1}{m} - \frac{1}{n}} & = \frac{\frac{n + m}{m n}}{\frac{n - m}{m n}} \\ = \frac{n + m}{n - m} \end{matrix}$
$\begin{matrix} \frac{x + \frac{x}{2}}{x - \frac{x}{4}} & = \frac{\frac{2 x + x}{2}}{\frac{4 x - x}{}} \\ = \frac{2 (3 x)}{3 x} \\ = 2 \end{matrix}$
$\begin{matrix} \frac{\frac{x}{y} - \frac{y}{x}}{1 + \frac{y}{x}} & = \frac{\frac{x^{2} - y^{2}}{x y}}{\frac{x + y}{x}} \\ = \frac{x^{2} - y^{2}}{y (x + y)} \\ = \frac{(x - y) (x + y)}{y (x + y)} \\ = \frac{x - y}{y} \end{matrix}$
$\begin{matrix} \frac{x + 4 + \frac{3}{x}}{x - 4 - \frac{5}{x}} & = \frac{\frac{x (x + 4) + 3}{x}}{\frac{x (x - 4) - 5}{x}} \\ = \frac{x^{2} + 4 x + 3}{x^{2} - 4 x - 5} \\ = \frac{(x + 1) (x + 3)}{(x + 5) (x + 1)} \\ = \frac{x + 3}{x + 5} \end{matrix}$
$\begin{matrix} \frac{a - 4 + \frac{4}{a}}{1 - \frac{2}{a}} & = \frac{\frac{a (a - 4) + 4}{a}}{\frac{a - 2}{a}} \\ = \frac{a^{2} - 4 a + 4}{a - 2} \\ = \frac{{(a - 2)}^{2}}{a - 2} \\ = a - 2 \end{matrix}$
$\begin{matrix} \frac{\frac{2 a^{2} - b^{2}}{a} - b}{\frac{4 a^{2} + b^{2}}{4 a b} + 1} & = \frac{\frac{2 a^{2} - b^{2} - a b}{a}}{\frac{4 a^{2} + b^{2} + 4 a b}{4 a b}} \\ = \frac{2 a^{2} - a b - b^{2}}{4 a^{2} + 4 a b + b^{2}} \\ = \frac{2 a^{2} - 2 a b + a b - b^{2}}{{(2 a + b)}^{2}} \\ = \frac{2 a (a - b) + b (a - b)}{{(2 a + b)}^{2}} \\ = \frac{(2 a + b) (a - b)}{{(2 a + b)}^{2}} \\ = \frac{a - b}{2 a + b} \end{matrix}$
$\begin{matrix} \frac{2 + \frac{3 a}{5 b}}{a + \frac{10 b}{3}} & = \frac{\frac{10 b + 3 a}{5 b}}{\frac{3 a + 10 b}{3}} \\ = \frac{3}{5 b} \end{matrix}$
$\begin{matrix} \frac{a - x + \frac{x^{2}}{a + x}}{a^{2} - \frac{a^{2}}{a + x}} & = \frac{\frac{(a - x) (a + x) + x^{2}}{a + x}}{\frac{a^{2} (a + x) - a^{2}}{a + x}} \\ = \frac{a^{2} - x^{2} + x^{2}}{a^{2} [(a + x) - 1]} \\ = \frac{a^{2}}{a^{2} (a + x - 1)} \\ = \frac{1}{a + x - 1} \end{matrix}$
$\begin{matrix} \frac{a + 5 - \frac{14}{a}}{1 + \frac{8}{a} + \frac{7}{a^{2}}} & = \frac{\frac{a (a + 5) - 14}{a}}{\frac{a^{2} + 8 a + 7}{a^{2}}} \\ = \frac{a (a^{2} + 5 a - 14)}{(a + 7) (a + 1)} \\ = \frac{a (a + 7) (a - 2)}{(a + 7) (a + 1)} \\ = \frac{a (a - 2)}{a + 1} \end{matrix}$
$\begin{matrix} \frac{\frac{1}{a} - \frac{9}{a^{2}} + \frac{20}{a^{3}}}{\frac{16}{a} - a} & = \frac{\frac{a^{2} - 9 a + 20}{a^{}}}{\frac{16 - a^{2}}{a}} \\ = \frac{a^{2} - 9 a + 20}{a^{2} (16 - a^{2})} \\ = \frac{(a - 5) (a - 4)}{a^{2} (4 - a) (4 + a)} \\ = - \frac{(a - 5) (a - 4)}{a^{2} (a - 4) (4 + a)} \\ = \frac{5 - a}{a^{2} (a + 4)} \end{matrix}$
$\begin{matrix} \frac{\frac{20 x^{2} + 7 x - 6}{x}}{\frac{4}{x^{2}} - 25} & = \frac{\frac{20 x^{2} + 15 x - 8 x - 6}{x}}{\frac{4 x^{2} - 25}{x^{2}}} \\ = \frac{x [5 x (4 x + 3) - 2 (4 x + 3)]}{4 x^{2} - 25} \\ = \frac{x (5 x - 2) (4 x + 3)}{(2 x - 5) (2 x + 5)} \\ = - \frac{x (2 x + 5) (4 x + 3)}{(2 x - 5) (2 x + 5)} \\ = \frac{x (4 x + 3)}{5 - 2 x} \end{matrix}$
$\begin{matrix} \frac{1 + \frac{1}{x - 1}}{1 + \frac{1}{x^{2} - 1}} & = \frac{\frac{x - 1 + 1}{x - 1}}{\frac{x^{2} - 1 + 1}{x^{2} - 1}} \\ = \frac{x (x^{2} - 1)}{x^{2} (x - 1)} \\ = \frac{(x + 1) (x - 1)}{x (x - 1)} \\ = \frac{x + 1}{x} \end{matrix}$
$\begin{matrix} \frac{a - \frac{a b}{a + b}}{a + \frac{a b}{a - b}} & = \frac{\frac{a (a + b) - a b}{a + b}}{\frac{a (a - b) + a b}{a - b}} \\ = \frac{\frac{a [(a + b) - b]}{a + b}}{\frac{a [(a - b) + b]}{a - b}} \\ = \frac{\frac{a (a + b - b)}{a + b}}{\frac{a (a - b + b)}{a - b}} \\ = \frac{a (a - b)}{a (a + b)} \\ = \frac{a - b}{a + b} \end{matrix}$
$\begin{matrix} \frac{x - 1 - \frac{5}{x + 3}}{x + 5 - \frac{35}{x + 3}} & = \frac{\frac{(x - 1) (x + 3) - 5}{x + 3}}{\frac{(x + 5) (x + 3) - 35}{x + 3}} \\ = \frac{x^{2} + 2 x - 3 - 5}{x^{2} + 8 x + 15 - 35} \\ = \frac{x^{2} + 2 x - 8}{x^{2} + 8 x - 20} \\ = \frac{(x + 4) (x - 2)}{(x + 10) (x - 2)} \\ = \frac{x + 4}{x + 10} \end{matrix}$
$\begin{matrix} \frac{a + 2 - \frac{7 a + 9}{a + 3}}{a - 4 + \frac{5 a - 11}{a + 1}} & = \frac{\frac{(a + 2) (a + 3) - (7 a + 9)}{a + 3}}{\frac{(a - 4) (a + 1) + 5 a - 11}{a + 1}} \\ = \frac{\frac{a^{2} + 5 a + 6 - 7 a - 9}{a + 3}}{\frac{a^{2} - 3 a - 4 + 5 a - 11}{a + 1}} \\ = \frac{\frac{a^{2} - 2 a - 3}{a + 3}}{\frac{a^{2} + 2 a - 15}{a + 1}} \\ = \frac{(a + 1) (a^{2} - 2 a - 3)}{(a + 3) (a^{2} + 2 a - 15)} \\ = \frac{(a + 1) (a - 3) (a + 1)}{(a + 3) (a + 5) (a - 3)} \\ = \frac{{(a + 1)}^{2}}{(a + 3) (a + 5)} \end{matrix}$