Question

Помогите решить пожалуйста.

$frac{x}{y} (frac{y}{x} - frac{x}{y} (frac{y^2}{x^2} - frac{x}{y} ( frac{y^3}{x^3} - frac{y^4}{x^4} )))$

Ответ должен выйти: $frac{x-y}{x}$ 

Answer 1

$frac{x}{y} *( frac{y}{x} - frac{x}{y} ( frac{y ^{2} }{ x^{2} } - frac{x}{y} ( frac{ y^{3} }{ x^{3} } - frac{y ^{4} }{ x^{4} } )))= frac{x-y}{y}$

$1) frac{y ^{3} }{ x^{3} } - frac{y ^{4} }{ x^{4} } = frac{xy ^{3}-y ^{4} }{ x^{4} } = frac{y ^{3} (x-y)}{ x^{4} } \ \ 2) frac{x}{y} * frac{y ^{3}(x-y) }{ x^{4} } = frac{y ^{2}(x-y) }{ x^{3} } = frac{xy ^{2} -y ^{3} }{ x^{3} } \ \ 3) frac{y ^{2} }{ x^{2} } - frac{xy ^{2}-y ^{3} }{ x^{3} } = frac{xy ^{2}-xy ^{2}+y ^{3}}{ x^{3} }= frac{y ^{3} }{ x^{2} } \ \ 4) frac{x}{y} * frac{y ^{3} }{ x^{3} } = frac{y ^{2} }{ x^{2} }$

$5) frac{y}{x} - frac{y ^{2} }{ x^{2} } = frac{xy-y ^{2} }{ x^{2} } = frac{y(x-y)}{ x^{2} } \ \ 6) frac{x}{y} * frac{y(x-y)}{ x^{2} } = frac{x-y}{x}$