6.2. Упростите выражение: а) \(\left(\frac{x}{y}-\frac{y}{x}\right)\cdot\frac{5xy}{x-y}\); б) \(\left(\frac{z^2}{t^2}+\frac{2z}{t}+1\right):\frac{t+z}{t}\); в) \(\left(\frac{a}{b}-\frac{b}{a}\right):\frac{a+b}{3ab}\); г) \(\left(1-\frac{2c}{d}+\frac{c^2}{d^2}\right)\cdot\frac{d}{c-d}\).

а)

\[\left(\frac{x}{y}-\frac{y}{x}\right)\cdot\frac{5xy}{x-y}=\frac{x^2-y^2}{xy}\cdot\frac{5xy}{x-y}=\frac{(x-y)(x+y)}{xy}\cdot\frac{5xy}{x-y}=5(x+y).\]

б)

\[\left(\frac{z^2}{t^2}+\frac{2z}{t}+1\right):\frac{t+z}{t}=\frac{(t+z)^2}{t^2}\cdot\frac{t}{t+z}=\frac{t+z}{t}.\]

в)

\[\left(\frac{a}{b}-\frac{b}{a}\right):\frac{a+b}{3ab}=\frac{a^2-b^2}{ab}\cdot\frac{3ab}{a+b}=\frac{(a-b)(a+b)}{ab}\cdot\frac{3ab}{a+b}=3(a-b).\]

г)

\[\left(1-\frac{2c}{d}+\frac{c^2}{d^2}\right)\cdot\frac{d}{c-d}=\frac{(d-c)^2}{d^2}\cdot\frac{d}{c-d}=\frac{c-d}{d}.\]

Ответ: а) \(5(x+y)\); б) \(\frac{t+z}{t}\); в) \(3(a-b)\); г) \(\frac{c-d}{d}\).