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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 132

Simplificar:

$\begin{matrix} \frac{2 a^{2}}{3 b} \times \frac{6 b^{2}}{4 a} & = \frac{2 a^{2}}{3 b} \times \frac{6 b^{2}}{4 a} \\ = a b \end{matrix}$
$\begin{matrix} \frac{x^{2} y}{5} \times \frac{10 a^{3}}{3 m^{2}} \times \frac{9 m}{x^{3}} & = \frac{x^{2} y}{5} \times \frac{a^{3}}{3 m^{2}} \times \frac{m}{x^{3}} \\ = \frac{6 a^{3} y}{m x} \end{matrix}$
$\begin{matrix} \frac{5 x^{2}}{7 y^{3}} \times \frac{4 y^{2}}{7 m^{3}} \times \frac{14 m}{5 x^{4}} & = \frac{5 x^{2}}{7 y^{3}} \times \frac{4 y^{2}}{7 m^{}} \times \frac{m}{5 x^{}} \\ = \frac{8}{7 m^{2} x^{2} y} \end{matrix}$
$\begin{matrix} \frac{5}{a} \times \frac{2 a}{b^{2}} \times \frac{3 b}{10} & = \frac{5}{a} \times \frac{2 a}{b^{2}} \times \frac{3 b}{10} \\ = \frac{3}{b} \end{matrix}$
$\begin{matrix} \frac{2 x^{3}}{15 a^{3}} \times \frac{3 a^{2}}{y} \times \frac{5 x^{2}}{7 x y^{2}} & = \frac{2 x^{3}}{15 a^{3}} \times \frac{3 a^{2}}{y} \times \frac{5 x^{2}}{7 x y^{2}} \\ = \frac{2 x^{4}}{7 a y^{3}} \end{matrix}$
$\begin{matrix} \frac{7 a}{6 m^{2}} \times \frac{3 m}{10 n^{2}} \times \frac{5 n^{4}}{14 a x} & = \frac{7 a}{m^{2}} \times \frac{3 m}{n^{2}} \times \frac{5 n^{}}{a x} \\ = \frac{n^{2}}{8 m x} \end{matrix}$
$\begin{matrix} \frac{2 x^{2} + x}{6} \times \frac{8}{4 x + 2} & = \frac{x (2 x + 1)}{} \times \frac{}{2 (2 x + 1)} \\ = \frac{2 x}{3} \end{matrix}$
$\begin{matrix} \frac{5 x + 25}{14} \times \frac{7 x + 7}{10 x + 50} & = \frac{5 (x + 5)}{} \times \frac{7 (x + 1)}{(x + 5)} \\ = \frac{x + 1}{4} \end{matrix}$
$\begin{matrix} \frac{m + n}{m n - n^{2}} \times \frac{n^{2}}{m^{2} - n^{2}} & = \frac{(m + n)}{n (m - n)} \times \frac{n^{2}}{(m - n) (m + n)} \\ = \frac{n}{{(m - n)}^{2}} \end{matrix}$
$\begin{matrix} \frac{x y - 2 y^{2}}{x^{2} + x y} \times \frac{x^{2} + 2 x y + y^{2}}{x^{2} - 2 x y} & = \frac{y (x - 2 y)}{x (x + y)} \times \frac{{(x + y)}^{2}}{x (x - 2 y)} \\ = \frac{y (x + y)}{x^{2}} \end{matrix}$
$\begin{matrix} \frac{x^{2} - 4 x y + 4 y^{2}}{x^{2} + 2 x y} \times \frac{x^{2}}{x^{2} - 4 y^{2}} & = \frac{{(x - 2 y)}^{2}}{x (x + 2 y)} \times \frac{x^{2}}{(x + 2 y) (x - 2 y)} \\ = \frac{x (x - 2 y)}{{(x + 2 y)}^{2}} \end{matrix}$
$\begin{matrix} \frac{2 x^{2} + 2 x}{2 x^{2}} \times \frac{x^{2} - 3 x}{x^{2} - 2 x - 3} & = \frac{2 x (x + 1)}{2 x^{2}} \times \frac{x (x - 3)}{(x - 3) (x + 1)} \\ = 1 \end{matrix}$
$\begin{matrix} \frac{a^{2} - a b + a - b}{a^{2} + 2 a + 1} \times \frac{3}{6 a^{2} - 6 a b} & = \frac{a (a - b) + (a - b)}{{(a + 1)}^{2}} \times \frac{3}{6 a (a - b)} \\ = \frac{(a - b) (a + 1)}{{(a + 1)}^{2}} \times \frac{3}{a (a - b)} \\ = \frac{1}{2 (a + 1)} \end{matrix}$
$\begin{matrix} \frac{{(x - y)}^{3}}{x^{3} - 1} \times \frac{x^{2} + x + 1}{{(x - y)}^{2}} & = \frac{{(x - y)}^{3}}{(x - 1) (x^{2} + x + 1)} \times \frac{x^{2} + x + 1}{{(x - y)}^{2}} \\ = \frac{x - y}{x - 1} \end{matrix}$
$\begin{matrix} \frac{2 a - 2}{2 a^{2} - 50} \times \frac{a^{2} - 4 a - 5}{3 a + 3} & = \frac{2 (a - 1)}{2 (a^{2} - 25)} \times \frac{(a - 5) (a + 1)}{3 (a + 1)} \\ = \frac{a - 1}{(a + 5) (a - 5)} \times \frac{a - 5}{3} \\ = \frac{a - 1}{3 (a + 5)} \end{matrix}$
$\begin{matrix} \frac{2 x^{2} - 3 x - 2}{6 x + 3} \times \frac{3 x + 6}{x^{2} - 4} & = \frac{2 x^{2} - 4 x + x - 2}{3 (2 x + 1)} \times \frac{3 (x + 2)}{(x - 2) (x + 2)} \\ = \frac{2 x (x - 2) + x - 2}{3 (2 x + 1)} \times \frac{3}{x - 2} \\ = \frac{(2 x + 1) (x - 2)}{3 (2 x + 1)} \times \frac{3}{(x - 2)} \\ = 1 \end{matrix}$
$\begin{matrix} \frac{y^{2} + 9 y + 18}{y - 5} \times \frac{5 y - 25}{5 y + 15} & = \frac{(y + 6) (y + 3)}{y - 5} \times \frac{5 (y - 5)}{5 (y + 3)} \\ = y + 6 \end{matrix}$
$\begin{matrix} \frac{x^{3} + 2 x^{2} - 3 x}{4 x^{2} + 8 x + 3} \times \frac{2 x^{2} + 3 x}{x^{2} - x} & = \frac{x (x^{2} + 2 x - 3)}{4 x^{2} + 2 x + 6 x + 3} \times \frac{x (2 x + 3)}{x (x - 1)} \\ = \frac{x (x + 3) (x - 1)}{2 x (2 x + 1) + 3 (2 x + 1)} \times \frac{2 x + 3}{x - 1} \\ = \frac{x (x + 3)}{(2 x + 3) (2 x + 1)} \times (2 x + 3) \\ = \frac{x (x + 3)}{2 x + 1} \end{matrix}$
$\begin{matrix} \frac{x^{3} - 27}{a^{3} - 1} \times \frac{a^{2} + a + 1}{x^{2} + 3 x + 9} & = \frac{(x - 3) (x^{2} + 3 x + 9)}{(a - 1) (a^{2} + a + 1)} \times \frac{a^{2} + a + 1}{x^{2} + 3 x + 9} \\ = \frac{x - 3}{a - 1} \end{matrix}$
$\begin{matrix} \frac{a^{2} + 4 a b + 4 b^{2}}{3} \times \frac{2 a + 4 b}{{(a + 2 b)}^{3}} & = \frac{{(a + 2 b)}^{2}}{3} \times \frac{2 (a + 2 b)}{{(a + 2 b)}^{3}} \\ = \frac{2}{3} \end{matrix}$
$\begin{matrix} \frac{1 - x}{a + 1} \times \frac{a^{2} + a}{x - x^{2}} \times \frac{x^{2}}{a} & = \frac{1 - x}{a + 1} \times \frac{a (a + 1)}{x (1 - x)} \times \frac{x^{2}}{a} \\ = x \end{matrix}$
$\begin{matrix} \frac{x^{2} + 2 x}{x^{2} - 16} \times \frac{x^{2} - 2 x - 8}{x^{3} + x^{2}} \times \frac{x^{2} + 4 x}{x^{2} + 4 x + 4} & = \frac{x (x + 2)}{(x - 4) (x + 4)} \times \frac{x^{2} - 4 x + 2 x - 8}{x^{2} (x + 1)} \times \frac{x (x + 4)}{{(x + 2)}^{2}} \\ = \frac{1}{x - 4} \times \frac{x (x - 4) + 2 (x - 4)}{x + 1} \times \frac{1}{x + 2} \\ = \frac{1}{x - 4} \times \frac{(x - 4) (x + 2)}{x + 1} \times \frac{1}{x + 2} \\ = \frac{1}{x + 1} \end{matrix}$
$\begin{matrix} \frac{{(m + n)}^{2} - x^{2}}{{(m + x)}^{2} - n^{2}} \times \frac{{(m - n)}^{2} - x^{2}}{m^{2} + m n - m x} & = \frac{[(m + n) - x] [(m + n) + x]}{[(m + x) - n] [(m + x) + n]} \times \frac{[(m - n) - x] [(m - n) + x]}{m (m + n - x)} \\ = \frac{[m + n - x] [m + n + x]}{[m + x - n] [m + x + n]} \times \frac{[m - n - x] [m - n + x]}{m (m + n - x)} \\ = \frac{m - n - x}{m} \end{matrix}$
$\begin{matrix} \frac{2 a^{3} + 2 a b^{2}}{2 a x^{2} - 2 a x} \times \frac{x^{3} - x}{a^{2} x + b^{2} x} \times \frac{x}{x + 1} & = \frac{2 a (a^{2} + b^{2})}{2 a x (x - 1)} \times \frac{x (x^{2} - 1)}{x (a^{2} + b^{2})} \times \frac{x}{x + 1} \\ = \frac{1}{x - 1} \times (x - 1) (x + 1) \times \frac{1}{x + 1} \\ = 1 \end{matrix}$
$\begin{matrix} \frac{a^{2} - 5 a + 6}{3 a - 15} \times \frac{6 a}{a^{2} - a - 30} \times \frac{a^{2} - 25}{2 a - 4} & = \frac{(a - 3) (a - 2)}{3 (a - 5)} \times \frac{6 a}{(a - 6) (a + 5)} \times \frac{(a + 5) (a - 5)}{2 (a - 2)} \\ = \frac{a (a - 3)}{a - 6} \end{matrix}$
$\begin{matrix} \frac{x^{2} - 3 x y - 10 y^{2}}{x^{2} - 2 x y - 8 y^{2}} \times \frac{x^{2} - 16 y^{2}}{x^{2} + 4 x y} \times \frac{x^{2} - 6 x y}{x + 2 y} & = \frac{(x - 5 y) (x + 2 y)}{(x - 4 y) (x + 2 y)} \times \frac{(x - 4 y) (x + 4 y)}{x (x + 4 y)} \times \frac{x (x - 6 y)}{x + 2 y} \\ = \frac{(x - 5 y) (x - 6 y)}{x + 2 y} \end{matrix}$
$\begin{matrix} \frac{x^{2} + 4 a x + 4 a^{2}}{3 a x - 6 a^{2}} \times \frac{2 a x - 4 a^{2}}{a x + a} \times \frac{6 a + 6 x}{x^{2} + 3 a x + 2 a^{2}} & = \frac{{(x + 2 a)}^{2}}{3 a (x - 2 a)} \times \frac{2 a (x - 2 a)}{a (x + 1)} \times \frac{(a + x)}{(x + 2 a) (x + a)} \\ = \frac{4 (x + 2 a)}{a (x + 1)} \end{matrix}$
$\begin{matrix} \frac{a^{2} - 81}{2 a^{2} + 10 a} \times \frac{a + 11}{a^{2} - 36} \times \frac{2 a - 12}{2 a + 18} \times \frac{a^{3} + 5 a^{2}}{2 a + 22} & = \frac{(a + 9) (a - 9)}{2 a (a + 5)} \times \frac{a + 11}{(a - 6) (a + 6)} \times \frac{2 (a - 6)}{2 (a + 9)} \times \frac{a^{2} (a + 5)}{2 (a + 11)} \\ = \frac{a (a - 9)}{4 (a + 6)} \end{matrix}$
$\begin{matrix} \frac{a^{2} + 7 a + 10}{a^{2} - 6 a - 7} \times \frac{a^{2} - 3 a - 4}{a^{2} + 2 a - 15} \times \frac{a^{3} - 2 a^{2} - 3 a}{a^{2} - 2 a - 8} & = \frac{(a + 5) (a + 2)}{(a - 7) (a + 1)} \times \frac{(a - 4) (a + 1)}{(a + 5) (a - 3)} \times \frac{a (a^{2} - 2 a - 3)}{(a - 4) (a + 2)} \\ = \frac{1}{a - 7} \times \frac{1}{a - 3} \times a (a - 3) (a + 1) \\ = \frac{a (a + 1)}{a - 7} \end{matrix}$
$\begin{matrix} \frac{x^{4} + 27 x}{x^{3} - x^{2} + x} \times \frac{x^{4} + x}{x^{4} - 3 x^{3} + 9 x^{2}} \times \frac{1}{x {(x + 3)}^{2}} \times \frac{x^{2}}{x - 3} & = \frac{x (x^{3} + 27)}{x (x^{2} - x + 1)} \times \frac{x (x^{3} + 1)}{x^{2} (x^{2} - 3 x + 9)} \times \frac{1}{x {(x + 3)}^{2}} \times \frac{x^{2}}{x - 3} \\ = \frac{(x + 3) (x^{2} - 3 x + 9)}{x^{2} - x + 1} \times \frac{(x + 1) (x^{2} - x + 1)}{x^{2} - 3 x + 9} \times \frac{1}{{(x + 3)}^{2}} \times \frac{1}{x - 3} \\ = \frac{x + 1}{(x + 3) (x - 3)} \\ = \frac{x + 1}{x^{2} - 9} \end{matrix}$