Đề bài

Thực hiện các phép tính sau:

ꦐa) \(\frac{{x - y}}{{xy}} + \frac{{y - z}}{{yz}} + \frac{{z - x}}{{z{\rm{x}}}}\)

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

Lời giải chi tiết

a)

🌼\(\begin{array}{*{20}{l}}{\frac{{x - y}}{{xy}} + \frac{{y - z}}{{yz}} + \frac{{z - x}}{{z{\rm{x}}}}}\\\begin{array}{l} = \frac{{z\left( {x - y} \right) + x\left( {y - z} \right) + y\left( {z - x} \right)}}{{xyz}}\\ = \frac{{z{\rm{x}} - zy + xy - x{\rm{z}} + yz - {\rm{yx}}}}{{xyz}}\\ = \frac{0}{{xyz}} = 0\end{array}\end{array}\)

b)

𝔉\(\begin{array}{l}\frac{x}{{{{\left( {x - y} \right)}^2}}} + \frac{y}{{{y^2} - {x^2}}} = \frac{x}{{{{\left( {x - y} \right)}^2}}} + \frac{{ - y}}{{{x^2} - {y^2}}}\\ = \frac{{x(x + y)}}{{{{\left( {x - y} \right)}^2}(x + y)}} + \frac{{ - y(x - y}}{{{{\left( {x - y} \right)}^2}\left( {x + y} \right)}}\\ = \frac{{{x^2} + xy - {\rm{yx}} + {y^2}}}{{{{\left( {x - y} \right)}^2}\left( {x + y} \right)}} = \frac{{{x^2} + {y^2}}}{{{{\left( {x - y} \right)}^2}\left( {x + y} \right)}}\end{array}\)