Упражнение 98

\({\largeа)}\ \frac{4}{y+2}-\frac{3}{y-2}+\frac{12}{y^2-4}=\frac{4}{y+2}-\frac{3}{y-2}+\frac{12}{ (y-2)(y+2)}=\frac{ 4(y-2)-3(y+2)+12}{ (y-2)(y+2)}=\frac{4y-8-3y-6+12}{ (y-2)(y+2)}=\frac{y-2}{ (y-2)(y+2)}=\frac{1}{y+2};\)

\({\largeб)}\ \frac{a}{a-6}-\frac{3}{a+6}+\frac{a^2}{36-a^2}=\frac{a}{a-6}-\frac{3}{a+6}-\frac{a^2}{a^2-36}=\frac{a}{a-6}-\frac{3}{a+6}-\frac{a^2}{ (a-6)(a+6)}=\frac{ a(a+6)-3(a-6)-a^2}{ (a-6)(a+6)}=\frac{a^2+6a-3a+18-a^2}{ (a-6)(a+6)}=\frac{3a+18}{ (a-6)(a+6)}=\frac{ 3(a+6)}{ (a-6)(a+6)}=\frac{3}{a-6};\)

\({\largeв)}\ \frac{x^2}{ (x-y)^2}-\frac{x+y}{2x-2y}=\frac{x^2}{ (x-y)^2}-\frac{x+y}{ 2(x-y)}=\frac{2x^2-(x+y)(x-y)}{ 2(x-y)^2}=\frac{2x^2-(x^2-y^2)}{ 2(x-y)^2}=\frac{2x^2-x^2+y^2}{ 2(x-y)^2}=\frac{x^2+y^2}{ 2(x-y)^2};\)

\({\largeг)}\ \frac{b}{ (a-b)^2}-\frac{a+b}{b^2-ab}=\frac{b}{ (a-b)^2}-\frac{a+b}{ b(b-a)}=\frac{b}{ (a-b)^2}+\frac{a+b}{ b(a-b)}=\frac{bb+(a+b)(a-b)}{ b(a-b)^2}=\frac{b^2+a^2-b^2}{ b(a-b)^2}=\frac{a^2}{ b(a-b)^2}.\)