Questions: Multiplicar: (2x-10y)/(x+6y) * (x+y)/(x^2-25y^2) Simplificar la respuesta.

Multiplicar: (2x-10y)/(x+6y) * (x+y)/(x^2-25y^2) Simplificar la respuesta.
Transcript text: Multiplicar: \[ \frac{2 x-10 y}{x+6 y} \cdot \frac{x+y}{x^{2}-25 y^{2}} \] Simplificar la respuesta.
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Solution

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To solve this problem, we need to multiply two rational expressions. First, multiply the numerators together and the denominators together. Then, simplify the resulting expression by factoring and canceling out common factors in the numerator and the denominator.

Paso 1: Multiplicar las expresiones racionales

Multiplicamos los numeradores y los denominadores de las dos expresiones racionales:

\[ \frac{2x - 10y}{x + 6y} \cdot \frac{x + y}{x^2 - 25y^2} = \frac{(2x - 10y)(x + y)}{(x + 6y)(x^2 - 25y^2)} \]

Paso 2: Simplificar la expresión resultante

El denominador \(x^2 - 25y^2\) es una diferencia de cuadrados, que se puede factorizar como \((x - 5y)(x + 5y)\). Así, la expresión se convierte en:

\[ \frac{(2x - 10y)(x + y)}{(x + 6y)(x - 5y)(x + 5y)} \]

Factorizamos el numerador \(2x - 10y\) como \(2(x - 5y)\):

\[ \frac{2(x - 5y)(x + y)}{(x + 6y)(x - 5y)(x + 5y)} \]

Cancelamos el factor común \((x - 5y)\) en el numerador y el denominador:

\[ \frac{2(x + y)}{(x + 6y)(x + 5y)} \]

Respuesta Final

La expresión simplificada es:

\[ \boxed{\frac{2(x + y)}{x^2 + 11xy + 30y^2}} \]

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