Giải bài 6.24 trang 19

Đề bài

Thực hiện các phép tính sau:

$a)\frac{x-y}{xy}+\frac{y-z}{yz}+\frac{z-x}{zx}$

$b)\frac{x}{{\left(x-y\right)}^2}+\frac{y}{y^2-x^2}$

Lời giải chi tiết

$\begin{array}{c}a)\frac{x-y}{xy}+\frac{y-z}{yz}+\frac{z-x}{zx}\\ =\frac{z\left(x-y\right)+x\left(y-z\right)+y\left(z-x\right)}{xyz}=\frac{zx-zy+xy-xz+yz-yx}{xyz}=0\end{array}$

$\begin{array}{c}b)\frac{x}{{\left(x-y\right)}^2}+\frac{y}{y^2-x^2}\\ =\frac{x}{{\left(x-y\right)}^2}-\frac{y}{x^2-y^2}\\ =\frac{x}{{\left(x-y\right)}^2}-\frac{y}{\left(x-y\right)\left(x+y\right)}\\ =\frac{x\left(x+y\right)-y\left(x-y\right)}{{\left(x-y\right)}^2\left(x+y\right)}\\ =\frac{x^2+xy-yx+y^2}{{\left(x-y\right)}^2\left(x+y\right)}=\frac{x^2+y^2}{{\left(x-y\right)}^2\left(x+y\right)}\end{array}$