$\frac{x -y\}{x^2-xy+y^2\}$ ÷ $\frac{x^3-y^3\}{x^3+y^3\}$ ÷ $\frac{x^2+y^2\}{x+y\}$

asked by guest
on Sep 20, 2024 at 10:43 pm



You asked:

Evaluate the expression: \(\frac{\frac{\frac{x - y}{{x}^{2} - x y + {y}^{2}}}{\frac{{x}^{3} - {y}^{3}}{{x}^{3} + {y}^{3}}}}{\frac{{x}^{2} + {y}^{2}}{x + y}}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{\frac{\frac{x - y}{{x}^{2} - x y + {y}^{2}}}{\frac{{x}^{3} - {y}^{3}}{{x}^{3} + {y}^{3}}}}{\frac{{x}^{2} + {y}^{2}}{x + y}} = \frac{\left(x^{3} + y^{3}\right) \left(x - y\right) \left(x + y\right)}{\left(x^{2} + y^{2}\right) \left(x^{3} - y^{3}\right) \left(x^{2} - x y + y^{2}\right)} \)

Expanded

\[\frac{\frac{\frac{x - y}{{x}^{2} - x y + {y}^{2}}}{\frac{{x}^{3} - {y}^{3}}{{x}^{3} + {y}^{3}}}}{\frac{{x}^{2} + {y}^{2}}{x + y}} = \frac{x}{\frac{x^{7}}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{x^{6} y}{x^{4} + x^{3} y + y^{3} x + y^{4}} + \frac{2 x^{5} y^{2}}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{2 x^{4} y^{3}}{x^{4} + x^{3} y + y^{3} x + y^{4}} + \frac{2 x^{3} y^{4}}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{2 x^{2} y^{5}}{x^{4} + x^{3} y + y^{3} x + y^{4}} + \frac{y^{6} x}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{y^{7}}{x^{4} + x^{3} y + y^{3} x + y^{4}}} - \frac{y}{\frac{x^{7}}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{x^{6} y}{x^{4} + x^{3} y + y^{3} x + y^{4}} + \frac{2 x^{5} y^{2}}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{2 x^{4} y^{3}}{x^{4} + x^{3} y + y^{3} x + y^{4}} + \frac{2 x^{3} y^{4}}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{2 x^{2} y^{5}}{x^{4} + x^{3} y + y^{3} x + y^{4}} + \frac{y^{6} x}{x^{4} + x^{3} y + y^{3} x + y^{4}} - \frac{y^{7}}{x^{4} + x^{3} y + y^{3} x + y^{4}}}\]

Factored

\[\frac{\frac{\frac{x - y}{{x}^{2} - x y + {y}^{2}}}{\frac{{x}^{3} - {y}^{3}}{{x}^{3} + {y}^{3}}}}{\frac{{x}^{2} + {y}^{2}}{x + y}} = \frac{\left(x + y\right)^{2}}{\left(x^{2} + y^{2}\right) \left(x^{2} + x y + y^{2}\right)}\]

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