Brüche mit Variablen rechnen: 1 / (x/y2-1/y) + 1/ (y/x2-1/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/y²-1/y)      +  1/    (y/x²-1/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/y² - 1/y) + 1/(y/x² - 1/x)

= 1/(x/y² - y/y²) + 1/(y/x² - x/x²)

= 1/((x - y)/y²) + 1/((y - x)/x²)

= y²/(x - y) + x²/(y - x)

= y²/(x - y) - x²/(x - y)

= (y² - x²)/(x - y)

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

= - (y + x)

= - (x + y)