\({\largeа)}\ ab+\frac{ab}{a+b}\left(\frac{a+b}{a-b}-a-b\right)=\frac{ab}{a-b}.\)
\(1)\ \frac{a+b}{a-b}-a-b=\frac{a+b-(a+b)(a-b)}{a-b}=\frac{ (a+b)(1-(a-b))}{a-b}=\frac{ (a+b)(1-a+b)}{a-b};\)
\(2)\ \frac{ab}{a+b}\cdot\frac{ (a+b)(1-a+b)}{a-b}=\frac{ ab(a+b)(1-a+b)}{ (a+b)(a-b)}=\frac{ ab(1-a+b)}{a-b}=\frac{ab-a^2b+ab^2}{a-b};\)
\(3)\ ab+\frac{ab-a^2b+ab^2}{a-b}=\frac{ ab(a-b)+ab-a^2b+ab^2}{a-b}=\frac{a^2b-ab^2+ab-a^2b+ab^2}{a-b}=\frac{ab}{a-b}.\)
\({\largeб)}\ \left(\frac{y^2-xy}{x^2+xy}-xy+y^2\right)\cdot\frac{x}{x-y}+\frac{y}{x+y}={-}xy.\)
\(1)\ \frac{y^2-xy}{x^2+xy}-xy+y^2=\frac{y^2-xy-(xy-y^2)(x^2+xy)}{x^2+xy}=\frac{{-}(xy-y^2)-(xy-y^2)(x^2+xy)}{x^2+xy}=\frac{ {-}(xy-y^2)(1+x^2+xy)}{ x(x+y)}=\frac{ {-}y(x-y)(1+x^2+xy)}{ x(x+y)};\)
\(2)\ \frac{ {-}y(x-y)(1+x^2+xy)}{ x(x+y)}\cdot\frac{x}{x-y}=\frac{ {-}y(x-y)(1+x^2+xy)\cdot{x}}{ x(x+y)(x-y)}=\frac{ {-}y(1+x^2+xy)}{x+y}=\frac{{-}y-x^2y-xy^2}{x+y};\)
\(3)\ \frac{{-}y-x^2y-xy^2}{x+y}+\frac{y}{x+y}=\frac{{-}y-x^2y-xy^2+y}{x+y}=\frac{{-}x^2y-xy^2}{x+y}=\frac{ {-}xy(x+y)}{x+y}={-}xy.\)
\({\largeв)}\ \left(\frac{1}{ (2a-b)^2}+\frac{2}{4a^2-b^2}+\frac{1}{ (2a+b)^2}\right)\cdot\frac{4a^2+4ab+b^2}{16a}=\frac{a}{ (2a-b)^2}.\)
\(1)\ \frac{1}{ (2a-b)^2}+\frac{2}{4a^2-b^2}+\frac{1}{ (2a+b)^2}=\frac{1}{ (2a-b)^2}+\frac{2}{ (2a-b)(2a+b)}+\frac{1}{ (2a+b)^2}=\frac{ (2a+b)^2+2(2a-b)(2a+b)+(2a-b)^2}{ (2a-b)^2(2a+b)^2}=\frac{4a^2+4ab+b^2+8a^2-2b^2+4a^2-4ab+b^2}{ (2a-b)^2(2a+b)^2}=\frac{16a^2}{ (2a-b)^2(2a+b)^2};\)
\(2)\ \frac{16a^2}{ (2a-b)^2(2a+b)^2}\cdot\frac{4a^2+4ab+b^2}{16a}=\frac{16a^2(2a+b)^2}{ 16a(2a-b)^2(2a+b)^2}=\frac{a}{ (2a-b)^2}.\)
\({\largeг)}\ \frac{4c^2}{ (c-2)^4}:\left(\frac{1}{ (c+2)^2}+\frac{1}{ (c-2)^2}+\frac{2}{c^2-4}\right)=\frac{ (c+2)^2}{ (c-2)^2}.\)
\(1)\ \frac{1}{ (c+2)^2}+\frac{1}{ (c-2)^2}+\frac{2}{c^2-4}=\frac{1}{ (c+2)^2}+\frac{1}{ (c-2)^2}+\frac{2}{ (c-2)(c+2)}=\frac{ (c-2)^2+(c+2)^2+2(c-2)(c+2)}{ (c-2)^2(c+2)^2}=\frac{c^2-4c+4+c^2+4c+4+2c^2-8}{ (c-2)^2(c+2)^2}=\frac{4c^2}{ (c-2)^2(c+2)^2};\)
\(2)\ \frac{4c^2}{ (c-2)^4}:\frac{4c^2}{ (c-2)^2(c+2)^2}=\frac{4c^2}{ (c-2)^4}\cdot\frac{ (c-2)^2(c+2)^2}{4c^2}=\frac{ 4c^2(c-2)^2(c+2)^2}{4c^2(c-2)^4}=\frac{ (c+2)^2}{ (c-2)^2}.\)
