Brüche mit Variablen rechnen: 1 / (x/y^2-1/y) + 1/ (y/x^2-1/x)

Question

Wie berechnet man diese Bruchrechnung Schritt für Schritt?

$$ \dfrac{ 1 }{ \dfrac{x}{y^2} - \dfrac{1}{y} } + \frac{ 1 }{ \dfrac{y}{x^2} - \dfrac{1}{x} } $$

1 /    (x/y^2-1/y)      +  1/    (y/x^2-1/x)

Answer 1

\(  \dfrac{ 1 }{ \dfrac{x}{y^2} - \dfrac{1}{y} } + \frac{ 1 }{ \dfrac{y}{x^2} - \dfrac{1}{x} } \\ = \dfrac{ 1 }{ \dfrac{x-y}{y^2} } + \frac{ 1 }{ \dfrac{y}{x^2} - \dfrac{y-x}{x^2} } \\ = \dfrac{ y^2 }{ x-y } + \dfrac{ x^2 }{ y - x } \\ = \dfrac{ y^2 }{ x-y } + \dfrac{ x^2 }{ -(x - y) } \\ = \dfrac{ 1 }{ x-y } · (y^2 - x^2) \\ = \dfrac{ -(x - y)·(x + y) }{ x-y }  \qquad | x-y ≠ 0 \\ = -(x + y) = -x - y \)

Answer 2

1/(x/y^2 - 1/y) + 1/(y/x^2 - 1/x)

= 1/(x/y^2 - y/y^2) + 1/(y/x^2 - x/x^2)

= 1/((x - y)/y^2) + 1/((y - x)/x^2)

= y^2/(x - y) + x^2/(y - x)

= y^2/(x - y) - x^2/(x - y)

= (y^2 - x^2)/(x - y)

= (y + x)(y - x)/(x - y)

= - (y + x)

= - (x + y)