Ejercicio 133 – Solucionario Baldor

Comparte esto 👍👍

DESCARGA

CAPITULO XIV

Operaciones con Fracciones

Ejercicio 133

Simplificar:

$\begin{matrix} (a + \frac{a}{b}) (a - \frac{a}{b + 1}) & = (\frac{a b + a}{b}) [\frac{a (b + 1) - a}{b + 1}] \\ = \frac{a (b + 1)}{b} (\frac{a b + a - a}{b + 1}) \\ = \frac{a^{2} b}{b} \\ = a^{2} \end{matrix}$
$\begin{matrix} (x - \frac{2}{x + 1}) (x + \frac{1}{x + 2}) & = [\frac{x (x + 1) - 2}{x + 1}] [\frac{x (x + 2) + 1}{x + 2}] \\ = [\frac{x^{2} + x - 2}{x + 1}] [\frac{x^{2} + 2 x + 1}{x + 2}] \\ = \frac{(x + 2) (x - 1)}{x + 1} \frac{{(x + 1)}^{2}}{x + 2} \\ = x^{2} - 1 \end{matrix}$
$\begin{matrix} (1 - \frac{x}{a + x}) (1 + \frac{x}{a}) & = [\frac{a + x - x}{a + x}] [\frac{a + x}{a}] \\ = \frac{a}{a} \\ = 1 \end{matrix}$
$\begin{matrix} (a + \frac{a b}{a - b}) (1 - \frac{b^{2}}{a^{2}}) & = [\frac{a (a - b) + a b}{a - b}] [\frac{a^{2} - b^{2}}{a^{2}}] \\ = [\frac{a^{2} - a b + a b}{a - b}] \frac{(a + b) (a - b)}{a^{2}} \\ = \frac{a^{2} (a + b)}{a^{2}} \\ = a + b \end{matrix}$
$\begin{matrix} (x + 2 - \frac{12}{x + 1}) (x - 2 + \frac{10 - 3 x}{x + 5}) & = [\frac{(x + 1) (x + 2) - 12}{x + 1}] [\frac{(x - 2) (x + 5) + 10 - 3 x}{x + 5}] \\ = [\frac{x^{2} + 3 x + 2 - 12}{x + 1}] [\frac{x^{2} + 3 x - 10 + 10 - 3 x}{x + 5}] \\ = [\frac{x^{2} + 3 x - 10}{x + 1}] (\frac{x^{2}}{x + 5}) \\ = \frac{(x + 5) (x - 2)}{x + 1} (\frac{x^{2}}{x + 5}) \\ = \frac{x^{2} (x - 2)}{x + 1} \end{matrix}$
$\begin{matrix} (1 + \frac{x}{y}) (x - \frac{x^{2}}{x + y}) & = (\frac{y + x}{y}) [\frac{x (x + y) - x^{2}}{x + y}] \\ = (\frac{1}{y}) [\frac{x^{2} + x y - x^{2}}{x + y}] \\ = \frac{x y}{y} \\ = x \end{matrix}$
$\begin{matrix} (a + x - \frac{a x + x^{2}}{a + 2 x}) (1 + \frac{x}{a + x}) & = [\frac{(a + x) (a + 2 x) - (a x + x^{2})}{a + 2 x}] [\frac{a + x + x}{a + x}] \\ = [\frac{a^{2} + 2 a x + a x + 2 x^{2} - a x - x^{2}}{a + 2 x}] [\frac{a + 2 x}{a + x}] \\ = (a^{2} + 2 a x + x^{2}) (\frac{1}{a + x}) \\ = {(a + x)}^{2} (\frac{1}{a + x}) \\ = a + x \end{matrix}$
$\begin{matrix} (x - \frac{x^{3} - 6 x}{x^{2} - 25}) (x + 1 - \frac{8}{x + 3}) & = [\frac{x (x^{2} - 25) - (x^{3} - 6 x)}{x^{2} - 25}] [\frac{(x + 1) (x + 3) - 8}{x + 3}] \\ = [\frac{x^{3} - 25 x - x^{3} + 6 x}{x^{2} - 25}] [\frac{x^{2} + 4 x + 3 - 8}{x + 3}] \\ = (\frac{- 19 x}{x^{2} - 25}) (\frac{x^{2} + 4 x - 5}{x + 3}) \\ = [\frac{- 19 x}{(x - 5) (x + 5)}] [\frac{(x + 5) (x - 1)}{x + 3}] \\ = \frac{19 x - 19 x^{2}}{(x - 5) (x + 3)} \end{matrix}$
$\begin{matrix} (m - \frac{m n}{m + n}) (1 + \frac{n^{3}}{m^{3}}) & = [\frac{m (m + n) - m n}{m + n}] (\frac{m^{3} + n^{3}}{m^{3}}) \\ = [\frac{m^{2} + m n - m n}{m + n}] [\frac{(m + n) (m^{2} - m n + n^{2})}{m^{3}}] \\ = m^{2} \frac{(m + n) (m^{2} - m n + n^{2})}{m^{3}} \\ = \frac{m^{2} - m n + n^{2}}{m} \end{matrix}$
$\begin{matrix} (a + 2 x - \frac{14 x^{2}}{2 a + x}) (a - x + \frac{a^{2} + 5 x^{2}}{a + 4 x}) & = [\frac{(a + 2 x) (2 a + x) - 14 x^{2}}{2 a + x}] [\frac{(a - x) (a + 4 x) + a^{2} + 5 x^{2}}{a + 4 x}] \\ = [\frac{2 a^{2} + a x + 4 a x + 2 x^{2} - 14 x^{2}}{2 a + x}] [\frac{a^{2} + 4 a x - a x - 4 x^{2} + a^{2} + 5 x^{2}}{a + 4 x}] \\ = [\frac{2 a^{2} + 5 a x - 12 x^{2}}{2 a + x}] [\frac{2 a^{2} + 3 a x + x^{2}}{a + 4 x}] \\ = [\frac{2 a^{2} - 3 a x + 8 a x - 12 x^{2}}{2 a + x}] [\frac{2 a^{2} + 2 a x + a x + x^{2}}{a + 4 x}] \\ = [\frac{a (2 a - 3 x) + 4 x (2 a - 3 x)}{2 a + x}] [\frac{2 a (a + x) + x (a + x)}{a + 4 x}] \\ = \frac{(a + 4 x) (2 a - 3 x)}{2 a + x} \times \frac{(2 a + x) (a + x)}{a + 4 x} \\ = (2 a - 3 x) (a + x) \end{matrix}$
$\begin{matrix} (1 + \frac{a}{b}) (1 - \frac{b}{a}) (1 + \frac{b^{2}}{a^{2} - b^{2}}) & = (\frac{b + a}{b}) (\frac{a - b}{a}) [\frac{a^{2} - b^{2} + b^{2}}{(a - b) (a + b)}] \\ = \frac{a^{2}}{a b} \\ = \frac{a}{b} \end{matrix}$
$\begin{matrix} (2 + \frac{2}{x + 1}) (3 - \frac{6}{x + 2}) (1 + \frac{1}{x}) & = [\frac{2 (x + 1) + 2}{x + 1}] [\frac{3 (x + 2) - 6}{x + 2}] (\frac{x + 1}{x}) \\ = (2 x + 4) [\frac{3 x + 6 - 6}{x + 2}] (\frac{1}{x}) \\ = 2 (x + 2) (\frac{3 x}{x + 2}) (\frac{1}{x}) \\ = 6 \end{matrix}$