\(1)\ \left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)=\frac{y+1}{y-1}\)
\(1)\ x+\frac{x}{y}={\frac{x}{1}}^{\backslash{y}}+{\frac{x}{y}}^{\backslash1}=\frac{xy+x}{y}\)
\(2)\ x-\frac{x}{y}={\frac{x}{1}}^{\backslash{y}}-{\frac{x}{y}}^{\backslash1}=\frac{xy-x}{y}\)
\(3)\ \frac{xy+x}{y}:\frac{xy-x}{y}=\frac{ x(y+1)}{y}\cdot\frac{y}{ x(y-1)}=\frac{y+1}{y-1}\)
\(2)\ \left(\frac{a}{b}+\frac{a+b}{a-b}\right)\cdot\frac{ab^2}{a^2+b^2}=\frac{ab}{a-b}\)
\(1)\ {\frac{a}{b}}^{\backslash{a\ -\ b}}+{\frac{a+b}{a-b}}^{\backslash{b}}=\frac{a^2-ab+ab+b^2}{ b(a-b)}=\frac{a^2+b^2}{ b(a-b)}\)
\(2)\ \frac{a^2+b^2}{ b(a-b)}\cdot\frac{ab^2}{a^2+b^2}=\frac{ab}{a-b}\)
\(3)\ \left(\frac{m}{m-1}-1\right):\frac{m}{mn-n}=\frac{n}{m}\)
\(1)\ \frac{m}{m-1}-1={\frac{m}{m-1}}^{\backslash1}-{\frac{1}{1}}^{\backslash{m\ -\ 1}}=\frac{m-(m-1)}{m-1}=\frac{m-m+1}{m-1}=\frac{1}{m-1}\)
\(2)\ \frac{1}{m-1}:\frac{m}{mn-n}=\frac{1}{m-1}\cdot\frac{ n(m-1)}{m}=\frac{n}{m}\)
\(4)\ \left(\frac{a}{b}-\frac{b}{a}\right)\cdot\frac{4ab}{a-b}=4a+4b\)
\(1)\ {\frac{a}{b}}^{\backslash{a}}-{\frac{b}{a}}^{\backslash{b}}=\frac{a^2-b^2}{ab}\)
\(2)\ \frac{a^2-b^2}{ab}\cdot\frac{4ab}{a-b}=\frac{ (a-b)(a+b)}{ab}\cdot\frac{4ab}{a-b}=4(a+b)=4a+4b\)
\(5)\ \frac{a}{b}-\frac{a^2-b^2}{b^2}:\frac{a+b}{b}=1\)
\(1)\ \frac{a^2-b^2}{b^2}:\frac{a+b}{b}=\frac{ (a-b)(a+b)}{b^2}\cdot\frac{b}{a+b}=\frac{a-b}{b}\)
\(2)\ {\frac{a}{b}}^{\backslash1}-{\frac{a-b}{b}}^{\backslash1}=\frac{a-(a-b)}{b}=\frac{a-a+b}{b}=\frac{b}{b}=1\)
\(6)\ \frac{7x}{x+2}-\frac{x-8}{3x+6}\cdot\frac{84}{x^2-8x}=\frac{7x-14}{x}\)
\(1)\ \frac{x-8}{3x+6}\cdot\frac{84}{x^2-8x}=\frac{x-8}{ 3(x+2)}\cdot\frac{84}{ x(x-8)}=\frac{28}{ x(x+2)}\)
\(2)\ {\frac{7x}{x+2}}^{\backslash{x}}-{\frac{28}{ x(x+2)}}^{\backslash1}=\frac{7x^2-28}{ x(x+2)}=\frac{ 7(x^2-4)}{ x(x+2)}=\frac{ 7(x-2)(x+2)}{ x(x+2)}=\frac{ 7(x-2)}{x}=\frac{7x-14}{x}\)
\(7)\ \left(a-\frac{9a-9}{a+3}\right):\frac{a^2-3a}{a+3}=\frac{a-3}{a}\)
\(1)\ a-\frac{9a-9}{a+3}={\frac{a}{1}}^{\backslash{a\ +\ 3}}-{\frac{9a-9}{a+3}}^{\backslash1}=\frac{a^2+3a-(9a-9)}{a+3}=\frac{a^2+3a-9a+9}{a+3}=\frac{a^2-6a+9}{a+3}\)
\(2)\ \frac{a^2-6a+9}{a+3}:\frac{a^2-3a}{a+3}=\frac{ (a-3)^2}{a+3}\cdot\frac{a+3}{ a(a-3)}=\frac{a-3}{a}\)
\(8)\ \left(\frac{a}{a+2}-\frac{8}{a+8}\right)\cdot\frac{a^2+8a}{a-4}=\frac{a^2+4a}{a+2}\)
\(1)\ {\frac{a}{a+2}}^{\backslash{a\ +\ 8}}-{\frac{8}{a+8}}^{\backslash{a\ +\ 2}}=\frac{a^2+8a-(8a+16)}{ (a+2)(a+8)}=\frac{a^2+8a-8a-16}{ (a+2)(a+8)}=\frac{a^2-16}{ (a+2)(a+8)}\)
\(2)\ \frac{a^2-16}{ (a+2)(a+8)}\cdot\frac{a^2+8a}{a-4}=\frac{ (a-4)(a+4)}{ (a+2)(a+8)}\cdot\frac{ a(a+8)}{a-4}=\frac{ a(a+4)}{a+2}=\frac{a^2+4a}{a+2}\)