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

а)

\[\left(\frac{c}{2}+\frac{c}{3}\right)\cdot\frac{1}{c^2}=\frac{5c}{6}\cdot\frac{1}{c^2}=\frac{5}{6c}.\]

б)

\[\left(\frac{2x}{y^2}-\frac{1}{2x}\right):\left(\frac{1}{y}+\frac{1}{2x}\right)=\frac{(2x-y)(2x+y)}{2xy^2}:\frac{2x+y}{2xy}=\frac{2x-y}{y}.\]

в)

\[\frac{d^2}{3}\cdot\left(\frac{d}{2}+\frac{2}{d^2}\right)=\frac{d^3}{6}+\frac{2}{3}=\frac{d^3+4}{6}.\]

г)

\[\left(\frac{a}{b^2}-\frac{1}{a}\right):\left(\frac{1}{b}+\frac{1}{a}\right)=\frac{(a-b)(a+b)}{ab^2}:\frac{a+b}{ab}=\frac{a-b}{b}.\]

Ответ: а) \(\frac{5}{6c}\); б) \(\frac{2x-y}{y}\); в) \(\frac{d^3+4}{6}\); г) \(\frac{a-b}{b}\).