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CAPITULO XIII

Simplificación de fracciones

Reducción de fracciones al mínimo común denominador

Ejercicio 125

Reducir al mínimo común denominador:

$\begin{matrix} \frac{a}{b}, \frac{1}{a b} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = a b \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ a b \div b = a & \frac{a}{b} = \frac{a \times a}{a b} = \frac{a^{2}}{a b} \\ a b \div a b = 1 & \frac{1}{a b} = \frac{1 \times 1}{a b} = \frac{1}{a b} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{a^{2}}{a b}, \frac{1}{a b} \end{matrix}$
$\begin{matrix} \frac{x}{2 a}, \frac{4}{3 a^{2} x} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 6 a^{2} x \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 6 a^{2} x \div 2 a = 3 a x & \frac{x}{2 a} = \frac{x \times 3 a x}{6 a^{2} x} = \frac{3 a x^{2}}{6 a^{2} x} \\ 6 a^{2} x \div 3 a^{2} x = 2 & \frac{4}{3 a^{2} x} = \frac{4 \times 2}{6 a^{2} x} = \frac{8}{6 a^{2} x} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 a x^{2}}{6 a^{2} x}, \frac{8}{6 a^{2} x} \end{matrix}$
$\begin{matrix} \frac{1}{2 x^{2}}, \frac{3}{4 x}, \frac{5}{8 x^{3}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 8 x^{3} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 8 x^{3} \div 2 x^{2} = 4 x & \frac{1}{2 x^{2}} = \frac{1 \times 4 x}{8 x^{3}} = \frac{4 x}{8 x^{3}} \\ 8 x^{3} \div 4 x = 2 x^{2} & \frac{3}{4 x} = \frac{3 \times 2 x^{2}}{8 x^{3}} = \frac{6 x^{2}}{8 x^{3}} \\ 8 x^{3} \div 8 x^{3} = 1 & \frac{5}{8 x^{3}} = \frac{5 \times 1}{8 x^{3}} = \frac{5}{8 x^{3}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{4 x}{8 x^{3}}, \frac{6 x^{2}}{8 x^{3}}, \frac{5}{8 x^{3}} \end{matrix}$
$\begin{matrix} \frac{3 x}{a b^{2}}, \frac{x}{a^{2} b}, \frac{3}{a^{3}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = a^{3} b^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ a^{3} b^{2} \div a b^{2} = a^{2} & \frac{3 x}{a b^{2}} = \frac{3 x \times a^{2}}{a^{3} b^{2}} = \frac{3 a^{2} x}{a^{3} b^{2}} \\ a^{3} b^{2} \div a^{2} b = a b & \frac{x}{a^{2} b} = \frac{x \times a b}{a^{3} b^{2}} = \frac{a b x}{a^{3} b^{2}} \\ a^{3} b^{2} \div a^{3} = b^{2} & \frac{3}{a^{3}} = \frac{3 \times b^{2}}{a^{3} b^{2}} = \frac{3 b^{2}}{a^{3} b^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 a^{2} x}{a^{3} b^{2}}, \frac{a b x}{a^{3} b^{2}}, \frac{3 b^{2}}{a^{3} b^{2}} \end{matrix}$
$\begin{matrix} \frac{7 y}{6 x^{2}}, \frac{1}{9 x y}, \frac{5 x}{12 y^{3}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 36 x^{2} y^{3} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 36 x^{2} y^{3} \div 6 x^{2} = 6 y^{3} & \frac{7 y}{6 x^{2}} = \frac{7 y \times 6 y^{3}}{36 x^{2} y^{3}} = \frac{42 y^{4}}{36 x^{2} y^{3}} \\ 36 x^{2} y^{3} \div 9 x y = 4 x y^{2} & \frac{1}{9 x y} = \frac{1 \times 4 x y^{2}}{36 x^{2} y^{3}} = \frac{4 x y^{2}}{36 x^{2} y^{3}} \\ 36 x^{2} y^{3} \div 12 y^{3} = 3 x^{2} & \frac{5 x}{12 y^{3}} = \frac{5 x \times 3 x^{2}}{36 x^{2} y^{3}} = \frac{15 x^{3}}{36 x^{2} y^{3}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{42 y^{4}}{36 x^{2} y^{3}}, \frac{4 x y^{2}}{36 x^{2} y^{3}}, \frac{15 x^{3}}{36 x^{2} y^{3}} \end{matrix}$
$\begin{matrix} \frac{a - 1}{3 a}, \frac{5}{6 a}, \frac{a + 2}{a^{2}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 6 a^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 6 a^{2} \div 3 a = 2 a & \frac{a - 1}{3 a} = \frac{(a - 1) \times 2 a}{6 a^{2}} = \frac{2 a^{2} - 2 a}{6 a^{2}} \\ 6 a^{2} \div 6 a = a & \frac{5}{6 a} = \frac{5 \times a}{6 a^{2}} = \frac{5 a}{6 a^{2}} \\ 6 a^{2} \div a^{2} = 6 & \frac{a + 2}{a^{2}} = \frac{(a + 2) \times 6}{6 a^{2}} = \frac{6 a + 12}{6 a^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{2 a^{2} - 2 a}{6 a^{2}}, \frac{5 a}{6 a^{2}}, \frac{6 a + 12}{6 a^{2}} \end{matrix}$
$\begin{matrix} \frac{x - y}{x^{2} y}, \frac{x + y}{3 x y^{2}}, 5 \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 3 x^{2} y^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 3 x^{2} y^{2} \div x^{2} y = 3 y & \frac{x - y}{x^{2} y} = \frac{(x - y) \times 3 y}{3 x^{2} y^{2}} = \frac{3 x y - 3 y^{2}}{3 x^{2} y^{2}} \\ 3 x^{2} y^{2} \div 3 x y^{2} = x & \frac{x + y}{3 x y^{2}} = \frac{(x + y) \times x}{3 x^{2} y^{2}} = \frac{x^{2} + x y}{3 x^{2} y^{2}} \\ 3 x^{2} y^{2} \div 1 = 3 x^{2} y^{2} & 5 = \frac{5 \times 3 x^{2} y^{2}}{3 x^{2} y^{2}} = \frac{15 x^{2} y^{2}}{3 x^{2} y^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 x y - 3 y^{2}}{3 x^{2} y^{2}}, \frac{x^{2} + x y}{3 x^{2} y^{2}}, \frac{15 x^{2} y^{2}}{3 x^{2} y^{2}} \end{matrix}$
$\begin{matrix} \frac{m + n}{2 m}, \frac{m - n}{5 m^{3} n}, \frac{1}{10 n^{2}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 10 m^{3} n^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 10 m^{3} n^{2} \div 2 m = 5 m^{2} n^{2} & \frac{m + n}{2 m} = \frac{(m + n) \times 5 m^{2} n^{2}}{10 m^{3} n^{2}} = \frac{5 m^{3} n^{2} + 5 m^{2} n^{3}}{10 m^{3} n^{2}} \\ 10 m^{3} n^{2} \div 5 m^{3} n = 2 n & \frac{m - n}{5 m^{3} n} = \frac{(m - n) \times 2 n}{10 m^{3} n^{2}} = \frac{2 m n - 2 n^{2}}{10 m^{3} n^{2}} \\ 10 m^{3} n^{2} \div 10 n^{2} = m^{3} & \frac{1}{10 n^{2}} = \frac{1 \times m^{3}}{10 m^{3} n^{2}} = \frac{m^{3}}{10 m^{3} n^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{5 m^{3} n^{2} + 5 m^{2} n^{3}}{10 m^{3} n^{2}}, \frac{2 m n - 2 n^{2}}{10 m^{3} n^{2}}, \frac{m^{3}}{10 m^{3} n^{2}} \end{matrix}$
$\begin{matrix} \frac{a + b}{6}, \frac{a - b}{2 a}, \frac{a^{2} + b^{2}}{3 b^{2}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 6 a b^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 6 a b^{2} \div 6 = a b^{2} & \frac{a + b}{6} = \frac{(a + b) \times a b^{2}}{6 a b^{2}} = \frac{a^{2} b^{2} + a b^{3}}{6 a b^{2}} \\ 6 a b^{2} \div 2 a = 3 b^{2} & \frac{a - b}{2 a} = \frac{(a - b) \times 3 b^{2}}{6 a b^{2}} = \frac{3 a b^{2} - 3 b^{3}}{6 a b^{2}} \\ 6 a b^{2} \div 3 b^{2} = 2 a & \frac{a^{2} + b^{2}}{3 b^{2}} = \frac{(a^{2} + b^{2}) \times 2 a}{6 a b^{2}} = \frac{2 a^{3} + 2 a b^{2}}{6 a b^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{a^{2} b^{2} + a b^{3}}{6 a b^{2}}, \frac{3 a b^{2} - 3 b^{3}}{6 a b^{2}}, \frac{2 a^{3} + 2 a b^{2}}{6 a b^{2}} \end{matrix}$
$\begin{matrix} \frac{2 a - b}{3 a^{2}}, \frac{3 b - a}{4 b^{2}}, \frac{a - 3 b}{2} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 12 a^{2} b^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 12 a^{2} b^{2} \div 3 a^{2} = 4 b^{2} & \frac{2 a - b}{3 a^{2}} = \frac{(2 a - b) \times 4 b^{2}}{12 a^{2} b^{2}} = \frac{8 a b^{2} - 4 b^{3}}{12 a^{2} b^{2}} \\ 12 a^{2} b^{2} \div 4 b^{2} = 3 a^{2} & \frac{3 b - a}{4 b^{2}} = \frac{(3 b - a) \times 3 a^{2}}{12 a^{2} b^{2}} = \frac{9 a^{2} b - 3 a^{3}}{12 a^{2} b^{2}} \\ 12 a^{2} b^{2} \div 2 = 6 a^{2} b^{2} & \frac{a - 3 b}{2} = \frac{(a - 3 b) \times 6 a^{2} b^{2}}{12 a^{2} b^{2}} = \frac{6 a^{3} b^{2} - 18 a^{2} b^{3}}{12 a^{2} b^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{8 a b^{2} - 4 b^{3}}{12 a^{2} b^{2}}, \frac{9 a^{2} b - 3 a^{3}}{12 a^{2} b^{2}}, \frac{6 a^{3} b^{2} - 18 a^{2} b^{3}}{12 a^{2} b^{2}} \end{matrix}$
$\begin{matrix} \frac{2}{5}, \frac{3}{x + 1} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 5 (x + 1) = 5 x + 5 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 5 (x + 1) \div 5 = x + 1 & \frac{2}{5} = \frac{2 \times (x + 1)}{5 x + 5} = \frac{2 x + 2}{5 x + 5} \\ 5 (x + 1) \div x + 1 = 5 & \frac{3}{x + 1} = \frac{3 \times 5}{5 x + 5} = \frac{15}{5 x + 5} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{2 x + 2}{5 x + 5}, \frac{15}{5 x + 5} \end{matrix}$
$\begin{matrix} \frac{a}{a + b}, \frac{b}{a^{2} - b^{2}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = a^{2} - b^{2} = (a + b) (a - b) \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (a + b) (a - b) \div a + b = a - b & \frac{a}{a + b} = \frac{a \times (a - b)}{a^{2} - b^{2}} = \frac{a^{2} - a b}{a^{2} - b^{2}} \\ (a + b) (a - b) \div a^{2} - b^{2} = 1 & \frac{b}{a^{2} - b^{2}} = \frac{b \times 1}{a^{2} - b^{2}} = \frac{b}{a^{2} - b^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{a^{2} - a b}{a^{2} - b^{2}}, \frac{b}{a^{2} - b^{2}} \end{matrix}$
$\begin{matrix} \frac{x}{x^{2} - 1}, \frac{1}{x^{2} - x - 2} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x^{2} - 1 = (x - 1) (x + 1) \\ x^{2} - x - 2 = (x - 2) (x + 1) \\ m . c . m = (x - 1) (x + 1) (x - 2) = (x^{2} - 1) (x - 2) = x^{3} - 2 x^{2} - x + 2 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (x - 1) (x + 1) (x - 2) \div x^{2} - 1 = x - 2 & \frac{x}{x^{2} - 1} = \frac{x \times (x - 2)}{x^{3} - 2 x^{2} - x + 2} = \frac{x^{2} - 2 x}{x^{3} - 2 x^{2} - x + 2} \\ (x - 1) (x + 1) (x - 2) \div x^{2} - x - 2 = x - 1 & \frac{1}{x^{2} - x - 2} = \frac{1 \times (x - 1)}{x^{3} - 2 x^{2} - x + 2} = \frac{x - 1}{x^{3} - 2 x^{2} - x + 2} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{x^{2} - 2 x}{x^{3} - 2 x^{2} - x + 2}, \frac{x - 1}{x^{3} - 2 x^{2} - x + 2} \end{matrix}$
$\begin{matrix} \frac{a - 3}{4 (a + 5)}, \frac{3 a}{8} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 8 (a + 5) = 8 a + 40 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 8 (a + 5) \div 4 (a + 5) = 2 & \frac{a - 3}{4 (a + 5)} = \frac{(a - 3) \times 2}{8 a + 40} = \frac{2 a - 6}{8 a + 40} \\ 8 (a + 5) \div 8 = a + 5 & \frac{3 a}{8} = \frac{3 a \times (a + 5)}{8 a + 40} = \frac{3 a^{2} + 15 a}{8 a + 40} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{2 a - 6}{8 a + 40}, \frac{3 a^{2} + 15 a}{8 a + 40} \end{matrix}$
$\begin{matrix} \frac{x^{2}}{3 (a - x)}, \frac{x}{6} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m . c . m = 6 (a - x) = 6 a - 6 x \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 6 (a - x) \div 3 (a - x) = 2 & \frac{x^{2}}{3 (a - x)} = \frac{x^{2} \times 2}{6 a - 6 x} = \frac{2 x^{2}}{6 a - 6 x} \\ 6 (a - x) \div 6 = a - x & \frac{x}{6} = \frac{x \times (a - x)}{6 a - 6 x} = \frac{a x - x^{2}}{6 a - 6 x} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{2 x^{2}}{6 a - 6 x}, \frac{a x - x^{2}}{6 a - 6 x} \end{matrix}$
$\begin{matrix} \frac{3}{x^{2}}, \frac{2}{x}, \frac{x + 3}{x^{2} - x} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x^{2} \\ x \\ x^{2} - x = x (x - 1) \\ m . c . m = x^{2} (x - 1) = x^{3} - x^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ x^{2} (x - 1) \div x^{2} = x - 1 & \frac{3}{x^{2}} = \frac{3 \times (x - 1)}{x^{3} - x^{2}} = \frac{3 x - 3}{x^{3} - x^{2}} \\ x^{2} (x - 1) \div x = x (x - 1) = x^{2} - x & \frac{2}{x} = \frac{2 \times (x^{2} - x)}{x^{3} - x^{2}} = \frac{2 x^{2} - 2 x}{x^{3} - x^{2}} \\ x^{2} (x - 1) \div x (x - 1) = x & \frac{x + 3}{x^{2} - x} = \frac{(x + 3) \times x}{x^{3} - x^{2}} = \frac{x^{2} + 3 x}{x^{3} - x^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 x - 3}{x^{3} - x^{2}}, \frac{2 x^{2} - 2 x}{x^{3} - x^{2}}, \frac{x^{2} + 3 x}{x^{3} - x^{2}} \end{matrix}$
$\begin{matrix} \frac{1}{2 a + 2 b}, \frac{a}{4 a - 4 b}, \frac{b}{8} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ 2 a + 2 b = 2 (a + b) \\ 4 a - 4 b = 4 (a - b) \\ 8 \\ m . c . m = 8 (a + b) (a - b) = 8 (a^{2} - b^{2}) = 8 a^{2} - 8 b^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 8 (a + b) (a - b) \div 2 (a + b) = 4 (a - b) = 4 a - 4 b & \frac{1}{2 a + 2 b} = \frac{1 \times (4 a - 4 b)}{8 a^{2} - 8 b^{2}} = \frac{4 a - 4 b}{8 a^{2} - 8 b^{2}} \\ 8 (a + b) (a - b) \div 4 (a - b) = 2 (a + b) = 2 a + 2 b & \frac{a}{4 a - 4 b} = \frac{a \times (2 a + 2 b)}{8 a^{2} - 8 b^{2}} = \frac{2 a^{2} + 2 a b}{8 a^{2} - 8 b^{2}} \\ 8 (a + b) (a - b) \div 8 = a^{2} - b^{2} & \frac{b}{8} = \frac{b \times (a^{2} - b^{2})}{8 a^{2} - 8 b^{2}} = \frac{a^{2} b - b^{3}}{8 a^{2} - 8 b^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{4 a - 4 b}{8 a^{2} - 8 b^{2}}, \frac{2 a^{2} + 2 a b}{8 a^{2} - 8 b^{2}}, \frac{a^{2} b - b^{3}}{8 a^{2} - 8 b^{2}} \end{matrix}$
$\begin{matrix} \frac{x}{x y}, \frac{y}{x^{2} + x y}, \frac{3}{x y + y^{2}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x y \\ x^{2} + x y = x (x + y) \\ x y + y^{2} = y (x + y) \\ m . c . m = x y (x + y) = x^{2} y + x y^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ x y (x + y) \div x y = x + y & \frac{x}{x y} = \frac{x \times (x + y)}{x^{2} y + x y^{2}} = \frac{x^{2} + x y}{x^{2} y + x y^{2}} \\ x y (x + y) \div x (x + y) = y & \frac{y}{x^{2} + x y} = \frac{y \times y}{x^{2} y + x y^{2}} = \frac{y^{2}}{x^{2} y + x y^{2}} \\ x y (x + y) \div y (x + y) = x & \frac{3}{x y + y^{2}} = \frac{3 \times x}{x^{2} y + x y^{2}} = \frac{3 x}{x^{2} y + x y^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{x^{2} + x y}{x^{2} y + x y^{2}}, \frac{y^{2}}{x^{2} y + x y^{2}}, \frac{3 x}{x^{2} y + x y^{2}} \end{matrix}$
$\begin{matrix} \frac{2}{a^{2} - b^{2}}, \frac{1}{a^{2} + a b}, \frac{a}{a^{2} - a b} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ a^{2} - b^{2} = (a - b) (a + b) \\ a^{2} + a b = a (a + b) \\ a^{2} - a b = a (a - b) \\ m . c . m = a (a - b) (a + b) = a (a^{2} - b^{2}) = a^{3} - a b^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ a (a - b) (a + b) \div (a - b) (a + b) = a & \frac{2}{a^{2} - b^{2}} = \frac{2 \times a}{a^{3} - a b^{2}} = \frac{2 a}{a^{3} - a b^{2}} \\ a (a - b) (a + b) \div a (a + b) = a - b & \frac{1}{a^{2} + a b} = \frac{1 \times (a - b)}{a^{3} - a b^{2}} = \frac{a - b}{a^{3} - a b^{2}} \\ a (a - b) (a + b) \div a (a - b) = a + b & \frac{a}{a^{2} - a b} = \frac{a \times (a + b)}{a^{3} - a b^{2}} = \frac{a^{2} + a b}{a^{3} - a b^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{2 a}{a^{3} - a b^{2}}, \frac{a - b}{a^{3} - a b^{2}}, \frac{a^{2} + a b}{a^{3} - a b^{2}} \end{matrix}$
$\begin{matrix} \frac{3 x}{x + 1}, \frac{x^{2}}{x - 1}, \frac{x^{3}}{x^{2} - 1} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x + 1 \\ x - 1 \\ x^{2} - 1 = (x - 1) (x + 1) \\ m . c . m = (x - 1) (x + 1) = x^{2} - 1 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (x - 1) (x + 1) \div (x + 1) = x - 1 & \frac{3 x}{x + 1} = \frac{3 x \times (x - 1)}{x^{2} - 1} = \frac{3 x^{2} - 3 x}{x^{2} - 1} \\ (x - 1) (x + 1) \div (x - 1) = x + 1 & \frac{x^{2}}{x - 1} = \frac{x^{2} \times (x + 1)}{x^{2} - 1} = \frac{x^{3} + x^{2}}{x^{2} - 1} \\ (x - 1) (x + 1) \div (x - 1) (x + 1) = 1 & \frac{x^{3}}{x^{2} - 1} = \frac{x^{3} \times 1}{x^{2} - 1} = \frac{x^{3}}{x^{2} - 1} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 x^{2} - 3 x}{x^{2} - 1}, \frac{x^{3} + x^{2}}{x^{2} - 1}, \frac{x^{3}}{x^{2} - 1} \end{matrix}$
$\begin{matrix} \frac{1}{m^{2} - n^{2}}, \frac{m}{m^{2} + m n}, \frac{n}{m^{2} - m n} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ m^{2} - n^{2} = (m - n) (m + n) \\ m^{2} + m n = m (m + n) \\ m^{2} - m n = m (m - n) \\ m . c . m = m (m - n) (m + n) = m (m^{2} - n^{2}) = m^{3} - m n^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ m (m - n) (m + n) \div (m - n) (m + n) = m & \frac{1}{m^{2} - n^{2}} = \frac{1 \times m}{m^{3} - m n^{2}} = \frac{m}{m^{3} - m n^{2}} \\ m (m - n) (m + n) \div m (m + n) = m - n & \frac{m}{m^{2} + m n} = \frac{m \times (m - n)}{m^{3} - m n^{2}} = \frac{m^{2} - m n}{m^{3} - m n^{2}} \\ m (m - n) (m + n) \div m (m - n) = m + n & \frac{n}{m^{2} - m n} = \frac{n \times (m + n)}{m^{3} - m n^{2}} = \frac{m n + n^{2}}{m^{3} - m n^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{m}{m^{3} - m n^{2}}, \frac{m^{2} - m n}{m^{3} - m n^{2}}, \frac{m n + n^{2}}{m^{3} - m n^{2}} \end{matrix}$
$\begin{matrix} \frac{n + 1}{n - 1}, \frac{n - 1}{n + 1}, \frac{n^{2} + 1}{n^{2} - 1} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ n - 1 \\ n + 1 \\ n^{2} - 1 = (n - 1) (n + 1) \\ m . c . m = (n - 1) (n + 1) = n^{2} - 1 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (n - 1) (n + 1) \div (n - 1) = n + 1 & \frac{n + 1}{n - 1} = \frac{(n + 1) \times (n + 1)}{n^{2} - 1} = \frac{{(n + 1)}^{2}}{n^{2} - 1} \\ (n - 1) (n + 1) \div (n + 1) = n - 1 & \frac{n - 1}{n + 1} = \frac{(n - 1) \times (n - 1)}{n^{2} - 1} = \frac{{(n - 1)}^{2}}{n^{2} - 1} \\ (n - 1) (n + 1) \div (n - 1) (n + 1) = 1 & \frac{n^{2} + 1}{n^{2} - 1} = \frac{(n^{2} + 1) \times 1}{n^{2} - 1} = \frac{n^{2} + 1}{n^{2} - 1} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{{(n + 1)}^{2}}{n^{2} - 1}, \frac{{(n - 1)}^{2}}{n^{2} - 1}, \frac{n^{2} + 1}{n^{2} - 1} \end{matrix}$
$\begin{matrix} \frac{a^{2} - b^{2}}{a^{2} + b^{2}}, \frac{a^{2} + b^{2}}{a^{2} - b^{2}}, \frac{a^{4} + b^{4}}{a^{4} - b^{4}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ a^{2} + b^{2} \\ a^{2} - b^{2} \\ a^{4} - b^{4} = (a^{2} - b^{2}) (a^{2} + b^{2}) \\ m . c . m = (a^{2} - b^{2}) (a^{2} + b^{2}) \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (a^{2} - b^{2}) (a^{2} + b^{2}) \div (a^{2} + b^{2}) = a^{2} - b^{2} & \frac{a^{2} - b^{2}}{a^{2} + b^{2}} = \frac{(a^{2} - b^{2}) \times (a^{2} - b^{2})}{a^{4} - b^{4}} = \frac{{(a^{2} - b^{2})}^{2}}{a^{4} - b^{4}} \\ (a^{2} - b^{2}) (a^{2} + b^{2}) \div (a^{2} - b^{2}) = a^{2} + b^{2} & \frac{a^{2} + b^{2}}{a^{2} - b^{2}} = \frac{(a^{2} + b^{2}) \times (a^{2} + b^{2})}{a^{4} - b^{4}} = \frac{{(a^{2} + b^{2})}^{2}}{a^{4} - b^{4}} \\ (a^{2} - b^{2}) (a^{2} + b^{2}) \div (a^{2} - b^{2}) (a^{2} + b^{2}) = 1 & \frac{a^{4} + b^{4}}{a^{4} - b^{4}} = \frac{(a^{4} + b^{4}) \times 1}{a^{4} - b^{4}} = \frac{a^{4} + b^{4}}{a^{4} - b^{4}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{{(a^{2} - b^{2})}^{2}}{a^{4} - b^{4}}, \frac{{(a^{2} + b^{2})}^{2}}{a^{4} - b^{4}}, \frac{a^{4} + b^{4}}{a^{4} - b^{4}} \end{matrix}$
$\begin{matrix} \frac{3 x}{x - 1}, \frac{x - 1}{x + 2}, \frac{1}{x^{2} + x - 2} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x - 1 \\ x + 2 \\ x^{2} + x - 2 = (x + 2) (x - 1) \\ m . c . m = (x + 2) (x - 1) = x^{2} + x - 2 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (x + 2) (x - 1) \div (x - 1) = x + 2 & \frac{3 x}{x - 1} = \frac{3 x \times (x + 2)}{x^{2} + x - 2} = \frac{3 x^{2} + 6 x}{x^{2} + x - 2} \\ (x + 2) (x - 1) \div (x + 2) = x - 1 & \frac{x - 1}{x + 2} = \frac{(x - 1) \times (x - 1)}{x^{2} + x - 2} = \frac{{(x - 1)}^{2}}{x^{2} + x - 2} \\ (x + 2) (x - 1) \div (x + 2) (x - 1) = 1 & \frac{1}{x^{2} + x - 2} = \frac{1 \times 1}{x^{2} + x - 2} = \frac{1}{x^{2} + x - 2} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 x^{2} + 6 x}{x^{2} + x - 2}, \frac{{(x - 1)}^{2}}{x^{2} + x - 2}, \frac{1}{x^{2} + x - 2} \end{matrix}$
$\begin{matrix} \frac{x}{2}, \frac{x}{5 x + 15}, \frac{x - 1}{10 x + 30} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ 2 \\ 5 x + 15 = 5 (x + 3) \\ 10 x + 30 = 10 (x + 3) \\ m . c . m = 10 (x + 3) = 10 x + 30 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 10 (x + 3) \div 2 = 5 (x + 3) = 5 x + 15 & \frac{x}{2} = \frac{x \times (5 x + 15)}{10 x + 30} = \frac{5 x^{2} + 15 x}{10 x + 30} \\ 10 (x + 3) \div 5 (x + 3) = 2 & \frac{x}{5 x + 15} = \frac{x \times 2}{10 x + 30} = \frac{2 x}{10 x + 30} \\ 10 (x + 3) \div 10 (x + 3) = 1 & \frac{x - 1}{10 x + 30} = \frac{(x - 1) \times 1}{10 x + 30} = \frac{x - 1}{10 x + 30} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{5 x^{2} + 15 x}{10 x + 30}, \frac{2 x}{10 x + 30}, \frac{x - 1}{10 x + 30} \end{matrix}$
$\begin{matrix} \frac{2 x - 1}{x + 4}, \frac{3 x + 1}{3 x + 12}, \frac{4 x + 3}{6 x + 24} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x + 4 \\ 3 x + 12 = 3 (x + 4) \\ 6 x + 24 = 6 (x + 4) \\ m . c . m = 6 (x + 4) = 6 x + 24 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 6 (x + 4) \div x + 4 = 6 & \frac{2 x - 1}{x + 4} = \frac{(2 x - 1) \times 6}{6 x + 24} = \frac{12 x - 6}{6 x + 24} \\ 6 (x + 4) \div 3 (x + 4) = 2 & \frac{3 x + 1}{3 x + 12} = \frac{(3 x + 1) \times 2}{6 x + 24} = \frac{6 x + 2}{6 x + 24} \\ 6 (x + 4) \div 6 (x + 4) = 1 & \frac{4 x + 3}{6 x + 24} = \frac{(4 x + 3) \times 1}{6 x + 24} = \frac{4 x + 3}{6 x + 24} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{12 x - 6}{6 x + 24}, \frac{6 x + 2}{6 x + 24}, \frac{4 x + 3}{6 x + 24} \end{matrix}$
$\begin{matrix} \frac{3}{a + 4}, \frac{2}{9 a^{2} - 25}, \frac{5}{3 a - 5} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ a + 4 \\ 9 a^{2} - 25 = (3 a - 5) (3 a + 5) \\ 3 a - 5 \\ m . c . m = (3 a - 5) (3 a + 5) (a + 4) = (9 a^{2} - 25) (a + 4) = 9 a^{3} + 36 a^{2} - 25 a - 100 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (3 a - 5) (3 a + 5) (a + 4) \div a + 4 = (3 a - 5) (3 a + 5) = (9 a^{2} - 25) & \frac{3}{a + 4} = \frac{3 \times (9 a^{2} - 25)}{9 a^{3} + 36 a^{2} - 25 a - 100} = \frac{27 a^{2} - 75}{9 a^{3} + 36 a^{2} - 25 a - 100} \\ (3 a - 5) (3 a + 5) (a + 4) \div (3 a - 5) (3 a + 5) = a + 4 & \frac{2}{9 a^{2} - 25} = \frac{2 \times (a + 4)}{9 a^{3} + 36 a^{2} - 25 a - 100} = \frac{2 a + 8}{9 a^{3} + 36 a^{2} - 25 a - 100} \\ (3 a - 5) (3 a + 5) (a + 4) \div (3 a - 5) = (3 a + 5) (a + 4) = 3 a^{2} + 17 a + 20 & \frac{5}{3 a - 5} = \frac{5 \times (3 a^{2} + 17 a + 20)}{9 a^{3} + 36 a^{2} - 25 a - 100} = \frac{15 a^{2} + 85 a + 100}{9 a^{3} + 36 a^{2} - 25 a - 100} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{27 a^{2} - 75}{9 a^{3} + 36 a^{2} - 25 a - 100}, \frac{2 a + 8}{9 a^{3} + 36 a^{2} - 25 a - 100}, \frac{15 a^{2} + 85 a + 100}{9 a^{3} + 36 a^{2} - 25 a - 100} \end{matrix}$
$\begin{matrix} \frac{x + 1}{x^{2} - 4}, \frac{x + 2}{x^{2} + x - 6}, \frac{3 x}{x^{2} + 5 x + 6} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x^{2} - 4 = (x + 2) (x - 2) \\ x^{2} + x - 6 = (x + 3) (x - 2) \\ x^{2} + 5 x + 6 = (x + 2) (x + 3) \\ m . c . m = (x + 2) (x - 2) (x + 3) = (x^{2} - 4) (x + 3) = x^{3} + 3 x^{2} - 4 x - 12 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (x + 2) (x - 2) (x + 3) \div (x + 2) (x - 2) (x + 3) = x + 3 & \frac{x + 1}{x^{2} - 4} = \frac{(x + 1) \times (x + 3)}{x^{3} + 3 x^{2} - 4 x - 12} = \frac{x^{2} + 4 x + 3}{x^{3} + 3 x^{2} - 4 x - 12} \\ (x + 2) (x - 2) (x + 3) \div (x - 2) (x + 3) = x + 2 & \frac{x + 2}{x^{2} + x - 6} = \frac{(x + 2) \times (x + 2)}{x^{3} + 3 x^{2} - 4 x - 12} = \frac{x^{2} + 4 x + 4}{x^{3} + 3 x^{2} - 4 x - 12} \\ (x + 2) (x - 2) (x + 3) \div (x + 2) (x + 3) = x - 2 & \frac{3 x}{x^{2} + 5 x + 6} = \frac{3 x \times (x - 2)}{x^{3} + 3 x^{2} - 4 x - 12} = \frac{3 x^{2} - 6 x}{x^{3} + 3 x^{2} - 4 x - 12} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{x^{2} + 4 x + 3}{x^{3} + 3 x^{2} - 4 x - 12}, \frac{x^{2} + 4 x + 4}{x^{3} + 3 x^{2} - 4 x - 12}, \frac{3 x^{2} - 6 x}{x^{3} + 3 x^{2} - 4 x - 12} \end{matrix}$
$\begin{matrix} \frac{a + 3}{a^{2} + a - 20}, \frac{5 a}{a^{2} - 7 a + 12}, \frac{a + 1}{a^{2} + 2 a - 15} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ a^{2} + a - 20 = (a + 5) (a - 4) \\ a^{2} - 7 a + 12 = (a - 4) (a - 3) \\ a^{2} + 2 a - 15 = (a + 5) (a - 3) \\ m . c . m = (a + 5) (a - 3) (a - 4) = (a^{2} + a - 20) (a - 3) = a^{3} - 2 a^{2} - 23 a + 60 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (a + 5) (a - 3) (a - 4) \div (a + 5) (a - 4) = a - 3 & \frac{a + 3}{a^{2} + a - 20} = \frac{(a + 3) \times (a - 3)}{a^{3} - 2 a^{2} - 23 a + 60} = \frac{a^{2} - 9}{a^{3} - 2 a^{2} - 23 a + 60} \\ (a + 5) (a - 3) (a - 4) \div (a - 3) (a - 4) = a + 5 & \frac{5 a}{a^{2} - 7 a + 12} = \frac{5 a \times (a + 5)}{a^{3} - 2 a^{2} - 23 a + 60} = \frac{5 a^{2} + 25 a}{a^{3} - 2 a^{2} - 23 a + 60} \\ (a + 5) (a - 3) (a - 4) \div (a + 5) (a - 3) = a - 4 & \frac{a + 1}{a^{2} + 2 a - 15} = \frac{(a + 1) \times (a - 4)}{a^{3} - 2 a^{2} - 23 a + 60} = \frac{a^{2} - 3 a - 4}{a^{3} - 2 a^{2} - 23 a + 60} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{a^{2} - 9}{a^{3} - 2 a^{2} - 23 a + 60}, \frac{5 a^{2} + 25 a}{a^{3} - 2 a^{2} - 23 a + 60}, \frac{a^{2} - 3 a - 4}{a^{3} - 2 a^{2} - 23 a + 60} \end{matrix}$
$\begin{matrix} \frac{a + 1}{a^{3} - 1}, \frac{2 a}{a^{2} + a + 1}, \frac{1}{a - 1} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ a^{3} - 1 = (a - 1) (a^{2} + a + 1) \\ a^{2} + a + 1 \\ a - 1 \\ m . c . m = (a - 1) (a^{2} + a + 1) = a^{3} - 1 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ (a - 1) (a^{2} + a + 1) \div (a - 1) (a^{2} + a + 1) = 1 & \frac{a + 1}{a^{3} - 1} = \frac{(a + 1) \times 1}{a^{3} - 1} = \frac{a + 1}{a^{3} - 1} \\ (a - 1) (a^{2} + a + 1) \div (a^{2} + a + 1) = a - 1 & \frac{2 a}{a^{2} + a + a} = \frac{2 a \times (a - 1)}{a^{3} - 1} = \frac{2 a^{2} - 2 a}{a^{3} - 1} \\ (a - 1) (a^{2} + a + 1) \div (a - 1) = a^{2} + a + 1 & \frac{1}{a - 1} = \frac{1 \times (a^{2} + a + 1)}{a^{3} - 1} = \frac{a^{2} + a + 1}{a^{3} - 1} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{a + 1}{a^{3} - 1}, \frac{2 a^{2} - 2 a}{a^{3} - 1}, \frac{a^{2} + a + 1}{a^{3} - 1} \end{matrix}$
$\begin{matrix} \frac{1}{x - 1}, \frac{1}{x^{3} - 1}, \frac{2}{3} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ x - 1 \\ x^{3} - 1 = (x - 1) (x^{2} + x + 1) \\ 3 \\ m . c . m = 3 (x - 1) (x^{2} + x + 1) = 3 (x^{3} - 1) = 3 x^{3} - 3 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 3 (x - 1) (x^{2} + x + 1) \div (x - 1) = 3 (x^{2} + x + 1) = 3 x^{2} + 3 x + 3 & \frac{1}{x - 1} = \frac{1 \times (3 x^{2} + 3 x + 3)}{3 x^{3} - 3} = \frac{3 x^{2} + 3 x + 3}{3 x^{3} - 3} \\ 3 (x - 1) (x^{2} + x + 1) \div (x - 1) (x^{2} + x + 1) = 3 & \frac{1}{x^{3} - 1} = \frac{1 \times 3}{3 x^{3} - 3} = \frac{3}{3 x^{3} - 3} \\ 3 (x - 1) (x^{2} + x + 1) \div 3 = (x - 1) (x^{2} + x + 1) = x^{3} - 1 & \frac{2}{3} = \frac{2 \times (x^{3} - 1)}{3 x^{3} - 3} = \frac{2 x^{3} - 2}{3 x^{3} - 3} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{3 x^{2} + 3 x + 3}{3 x^{3} - 3}, \frac{3}{3 x^{3} - 3}, \frac{2 x^{3} - 2}{3 x^{3} - 3} \end{matrix}$
$\begin{matrix} \frac{3}{2 a^{2} + 2 a b}, \frac{b}{a^{2} x + a b x}, \frac{1}{4 a x^{2} - 4 b x^{2}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ 2 a^{2} + 2 a b = 2 a (a + b) \\ a^{2} x + a b x = a x (a + b) \\ 4 a x^{2} - 4 b x^{2} = 4 x^{2} (a - b) \\ m . c . m = 4 a x^{2} (a - b) (a + b) = 4 a x^{2} (a^{2} - b^{2}) = 4 a^{3} x^{2} - 4 a b^{2} x^{2} \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 4 a x^{2} (a - b) (a + b) \div 2 a (a + b) = 2 x^{2} (a - b) = 2 a x^{2} - 2 b x^{2} & \frac{3}{2 a^{2} + 2 a b} = \frac{3 \times (2 a x^{2} - 2 b x^{2})}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} = \frac{6 a x^{2} - 6 b x^{2}}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} \\ 4 a x^{2} (a - b) (a + b) \div a x (a + b) = 4 x (a - b) = 4 a x - 4 b x & \frac{b}{a^{2} x + a b x} = \frac{b \times (4 a x - 4 b x)}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} = \frac{4 a b x - 4 b^{2} x}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} \\ 4 a x^{2} (a - b) (a + b) \div 4 x^{2} (a - b) = a (a + b) = a^{2} + a b & \frac{1}{4 a x^{2} - 4 b x^{2}} = \frac{1 \times (a^{2} + a b)}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} = \frac{a^{2} + a b}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{6 a x^{2} - 6 b x^{2}}{4 a^{3} x^{2} - 4 a b^{2} x^{2}}, \frac{4 a b x - 4 b^{2} x}{4 a^{3} x^{2} - 4 a b^{2} x^{2}}, \frac{a^{2} + a b}{4 a^{3} x^{2} - 4 a b^{2} x^{2}} \end{matrix}$
$\begin{matrix} \frac{1}{a - 1}, \frac{a + 1}{{(a - 1)}^{2}}, \frac{3 (a + 1)}{{(a - 1)}^{3}} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ a - 1 \\ {(a - 1)}^{2} \\ {(a - 1)}^{3} \\ m . c . m = {(a - 1)}^{3} = a^{3} - 3 a^{2} + 3 a - 1 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ {(a - 1)}^{3} \div a - 1 = {(a - 1)}^{2} = a^{2} - 2 a + 1 & \frac{1}{a - 1} = \frac{1 \times (a^{2} - 2 a + 1)}{a^{3} - 3 a^{2} + 3 a - 1} = \frac{a^{2} - 2 a + 1}{a^{3} - 3 a^{2} + 3 a - 1} \\ {(a - 1)}^{3} \div {(a - 1)}^{2} = a - 1 & \frac{a + 1}{{(a - 1)}^{2}} = \frac{(a + 1) \times (a - 1)}{a^{3} - 3 a^{2} + 3 a - 1} = \frac{a^{2} - 1}{a^{3} - 3 a^{2} + 3 a - 1} \\ {(a - 1)}^{3} \div {(a - 1)}^{3} = 1 & \frac{3 (a + 1)}{{(a - 1)}^{3}} = \frac{(3 a + 3) \times 1}{a^{3} - 3 a^{2} + 3 a - 1} = \frac{3 a + 3}{a^{3} - 3 a^{2} + 3 a - 1} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{a^{2} - 2 a + 1}{a^{3} - 3 a^{2} + 3 a - 1}, \frac{a^{2} - 1}{a^{3} - 3 a^{2} + 3 a - 1}, \frac{3 a + 3}{a^{3} - 3 a^{2} + 3 a - 1} \end{matrix}$
$\begin{matrix} \frac{2 x - 3}{6 x^{2} + 7 x + 2}, \frac{3}{2 x + 1}, \frac{2 x - 1}{6 x + 4} \\ O b t e n e m o s e l m . c . m d e l o s d e n o m i n a d o r e s \\ 6 x^{2} + 7 x + 2 = 6 x^{2} + 3 x + 4 x + 2 = 3 x (2 x + 1) + 2 (2 x + 1) = (2 x + 1) (3 x + 2) \\ 2 x + 1 \\ 6 x + 4 = 2 (3 x + 2) \\ m . c . m = 2 (2 x + 1) (3 x + 2) = 2 (6 x^{2} + 7 x + 2) = 12 x^{2} + 14 x + 4 \\ D i v i d i m o s e l m . c . m p a r a l o s d e n o m i n a d o r e s \\ y e s t e r e s u l t a d o m u l t i p l i c a m o s p a r a l o s n u m e r a d o r e s \\ 2 (2 x + 1) (3 x + 2) \div (2 x + 1) (3 x + 2) = 2 & \frac{2 x - 3}{6 x^{2} + 7 x + 2} = \frac{2 \times (2 x - 3)}{12 x^{2} + 14 x + 4} = \frac{4 x - 6}{12 x^{2} + 14 x + 4} \\ 2 (2 x + 1) (3 x + 2) \div (2 x + 1) = 2 (3 x + 2) = 6 x + 4 & \frac{3}{2 x + 1} = \frac{3 \times (6 x + 4)}{12 x^{2} + 14 x + 4} = \frac{18 x + 12}{12 x^{2} + 14 x + 4} \\ 2 (2 x + 1) (3 x + 2) \div 2 (3 x + 2) = 2 x + 1 & \frac{2 x - 1}{6 x + 4} = \frac{(2 x - 1) \times (2 x - 1)}{12 x^{2} + 14 x + 4} = \frac{4 x^{2} - 1}{12 x^{2} + 14 x + 4} \\ F r a c c i o n e s r e d u c i d a s a l m í n i m o c o m ú n d e n o m i n a d o r \\ \frac{4 x - 6}{12 x^{2} + 14 x + 4}, \frac{18 x + 12}{12 x^{2} + 14 x + 4}, \frac{4 x^{2} - 1}{12 x^{2} + 14 x + 4} \end{matrix}$