[(x+y)/(x-y)]/ [( x2x^{2} + xy + y2y^{2} ) / x3x^{3}-y3y^{3} ]

asked by guest
on Mar 25, 2025 at 6:34 am



You asked:

Evaluate the expression: x+yx−yx2+xy+y2x3−y3\frac{\frac{x + y}{x - y}}{\frac{{x}^{2} + x y + {y}^{2}}{{x}^{3}} - {y}^{3}}

MathBot Answer:

Evaluated



x+yx−yx2+xy+y2x3−y3=x+y(−y3+x2+xy+y2x3)(x−y)\displaystyle \frac{\frac{x + y}{x - y}}{\frac{{x}^{2} + x y + {y}^{2}}{{x}^{3}} - {y}^{3}} = \frac{x + y}{\left(- y^{3} + \frac{x^{2} + x y + y^{2}}{x^{3}}\right) \left(x - y\right)}

Expanded

x+yx−yx2+xy+y2x3−y3=x−y3x+y4+1−y3x3+y−y3x+y4+1−y3x3\frac{\frac{x + y}{x - y}}{\frac{{x}^{2} + x y + {y}^{2}}{{x}^{3}} - {y}^{3}} = \frac{x}{- y^{3} x + y^{4} + 1 - \frac{y^{3}}{x^{3}}} + \frac{y}{- y^{3} x + y^{4} + 1 - \frac{y^{3}}{x^{3}}}

Factored

x+yx−yx2+xy+y2x3−y3=−x3(x+y)(x−y)(x3y3−x2−xy−y2)\frac{\frac{x + y}{x - y}}{\frac{{x}^{2} + x y + {y}^{2}}{{x}^{3}} - {y}^{3}} = - \frac{x^{3} \left(x + y\right)}{\left(x - y\right) \left(x^{3} y^{3} - x^{2} - x y - y^{2}\right)}

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