Задача 72250 Упростить:...

62aaac79225ec1501a9067b6

15.07.2023 10:50:48

Упростить: [m](\frac{1}{x}+\frac{1}{y})[(x-y)^2+xy]+(\frac{1}{x}-\frac{1}{y})[(x+y)^2-xy][/m]

5f3ea7e3faf909182968ddd9

15.07.2023 12:28:52

★

[m](\frac{1}{x}+\frac{1}{y})[(x−y)2+xy]+(\frac{1}{x}-\frac{1}{y})[(x+y)2−xy]=(\frac{y+x}{xy})[x^2-2xy+y^2+xy]+(\frac{y-x}{xy})[x^2+2xy+y^2−xy]=[/m]


[m]=\frac{(y+x)(x^2-xy+y^2)}{xy}+\frac{(y-x)(x^2+xy+y^2)}{xy}=\frac{y^3+x^3}{xy}+\frac{y^3-x^3}{xy}=\frac{y^3+x^3+y^3-x^3}{xy}=\frac{2y^2}{x}[/m]

626fab9a354ab65fb2f43abb

15.07.2023 21:54:26

Простое выражение.
[m](\frac{1}{x}+\frac{1}{y})[(x-y)^2+xy] + (\frac{1}{x}-\frac{1}{y})[(x+y)^2-xy] = [/m]
[m] =(\frac{1}{x}+\frac{1}{y})(x^2-2xy+y^2+xy) + (\frac{1}{x}-\frac{1}{y})(x^2+2xy+y^2-xy) =[/m]
[m]=(\frac{1}{x}+\frac{1}{y})(x^2-xy+y^2) + (\frac{1}{x}-\frac{1}{y})(x^2+xy+y^2) =[/m]
[m] =(x+\frac{x^2}{y}-y-x+\frac{y^2}{x}+y) + (x-\frac{x^2}{y}+y-x+\frac{y^2}{x}-y) =[/m]
[m]=\frac{x^2}{y}+\frac{y^2}{x} - \frac{x^2}{y}+\frac{y^2}{x} =\frac{2y^2}{x}[/m]