Задание 537

Решение:

• \({\largeа)}\ \frac{x+y}{x}-\frac{x}{x-y}+\frac{y^2}{x^2-xy}=\frac{x+y^{\backslash{x\ -\ y}}}{x\phantom{^{\backslash{x\ -\ y}}}}-\frac{x^{\backslash{x}}}{x-y\phantom{^{\backslash{x}}}}+\frac{y^{2\backslash1}}{ x(x-y)\phantom{^{\backslash1}}}=\frac{ (x+y)(x-y)}{ x(x-y)}-\frac{x\cdot{x}}{ x(x-y)}+\frac{y^2\cdot1}{ x(x-y)}=\frac{x^2-y^2-x^2+y^2}{ x(x-y)}=\frac{0}{ x(x-y)}=0\)

• \({\largeб)}\ \frac{1}{m+2}+\frac{1}{m-2}-\frac{4}{m^2-4}=\frac{1^{\backslash{m\ -\ 2}}}{m+2\phantom{^{\backslash{m\ -\ 2}}}}+\frac{1^{\backslash{m\ +\ 2}}}{m-2\phantom{^{\backslash{m\ +\ 2}}}}-\frac{4^{\backslash1}}{ (m+2)(m-2)\phantom{^{\backslash1}}}=\frac{ 1\cdot(m-2)}{ (m+2)(m-2)}+\frac{ 1\cdot(m+2)}{ (m+2)(m-2)}-\frac{4\cdot1}{ (m+2)(m-2)}=\frac{m-2+m+2-4}{ (m+2)(m-2)}=\frac{2m-4}{ (m+2)(m-2)}=\frac{ 2(m-2)}{ (m+2)(m-2)}=\frac{2}{m+2}\)

• \({\largeв)}\ \frac{3x^2+3xy}{4xy+6ay}\cdot\left(\frac{x}{ax+ay}+\frac{3}{2x+2y}\right)=\frac{ 3x(x+y)}{ 2y(2x+3a)}\cdot\left(\frac{x^{\backslash2}}{ a(x+y)\phantom{^{\backslash2}}}+\frac{3^{\backslash{a}}}{ 2(x+y)\phantom{^{\backslash{a}}}}\right)=\frac{ 3x(x+y)}{ 2y(2x+3a)}\cdot\frac{2x+3a}{ 2a(x+y)}=\frac{3x}{2y}\cdot\frac{1}{2a}=\frac{3x}{4ay}\)

• \({\largeг)}\ \left(\frac{c-d}{c^2+cd}-\frac{c}{d^2+cd}\right):\left(\frac{d^2}{c^3-cd^2}+\frac{1}{c+d}\right)=\left(\frac{c-d}{ c(c+d)}-\frac{c}{ d(d+c)}\right):\left(\frac{d^2}{ c(c^2-d^2)}+\frac{1}{c+d}\right)=\left(\frac{c-d^{\backslash{d}}}{ c(c+d)\phantom{^{\backslash{d}}}}-\frac{c^{\backslash{c}}}{ d(d+c)\phantom{^{\backslash{c}}}}\right):\left(\frac{d^{2\backslash1}}{ c(c+d)(c-d)\phantom{^{\backslash1}}}+\frac{1^{\backslash{ c(c\ -\ d)}}}{c+d\phantom{^{\backslash{ c(c\ -\ d)}}}}\right)=\frac{ (c-d)\cdot{d}-c\cdot{c}}{ cd(c+d)}:\frac{ d^2\cdot1+1\cdot{c}(c-d)}{ c(c+d)(c-d)}=\frac{cd-d^2-c^2}{ cd(c+d)}\cdot\frac{ c(c+d)(c-d)}{d^2+c^2-cd}=\frac{ {-}(d^2+c^2-cd)}{d}\cdot\frac{c-d}{d^2+c^2-cd}=\frac{ {-}(c-d)}{d}=\frac{d-c}{d}\)