Решение №1855: \(\frac{9n}{m(3n-m)}+\frac{m}{n(m-3n)}=\frac{9n}{m(3n-m)}-\frac{m}{n(3n-m)}=\frac{9n^{2}-m^{2}}{mn(3n-m)}=\frac{(3n-m)(3n+m)}{mn(3n-m)}=\frac{3n+m}{mn}; m \neq 0, n \neq 0; 3n-m \neq 0, m \neq 3n\)

Ответ: \(m \neq 3n\)

Решение №1858: \(\frac{m}{m-n}-\frac{n}{m+n}=\frac{m(m+n)-n(m-n)}{(m-n)(m+n)}=\frac{m^{2}+mn-mn+n^{2}}{m^{2}-n^{2}}=\frac{m^{2}+n^{2}}{m^{2}-n^{2}}\)

Ответ: \(\frac{m^{2}+n^{2}}{m^{2}-n^{2}}\)

Решение №1862: \(\frac{x+y}{4x(x-y)}-\frac{x-y}{4x(x+y)}=\frac{(x+y)(x+y)-(x-y)(x-y)}{4x(x-y)(x+y)}=\frac{(x+y)^{2}-(x-y)^{2}}{4x(x^{2}-y^{2})}=\frac{x^{2}+2xy+y^{2}-(x^{2}-2xy+y^{2})}{4x(x^{2}-y^{2}}=\frac{x^{2}+2xy+y^{2}-x^{2}+2xy-y^{2}}{4x(x^{2}-y^{2})}=\frac{4xy}{4x(x^{2}-y^{2}}=\frac{y}{x^{2}-y^{2}}\)

Ответ: \(\frac{y}{x^{2}-y^{2}}\)

Решение №1868: \(\frac{-6x-3}{(2x-3)(2x+3)}-\frac{2}{3-2x}=\frac{-3(2x+3)}{(2x-3)(2x+3)}+\frac{2}{2x-3}=-\frac{3}{2x-3}+\frac{2}{2x-3}=\frac{-3+2}{2x-3}=\frac{-1}{3-2x}=\frac{1}{3-2x}\)

Ответ: \(\frac{1}{3-2x}\)

Решение №1875: \(\frac{a+5}{(a-3)(a+3)}+\frac{a+4}{a(-a-3)}=\frac{a+5}{(a-3)(a+3)}-\frac{a+4}{a(a+3)}=\frac{a(a+5)-(a+4)(a-3)}{a(a-3)(a+3)}=\frac{a^2}+5a-(a^{2}-3a+4a-12)}{a(a-3)(a+3)}=\frac{a^{2}+5a-a^{2}-a+12}{a(a-3)(a+3)}=\frac{4a+12}{a(a-3)(a+3)}=\frac{4(a+3)}{a(a-3)(a+3)}=\frac{a^{2}+5a-a^{2}-a+12}{a(a-3)(a+3)}=\frac{4a+12}{a(a-3)(a+3)}=\frac{4(a+3)}{a(a-3)(a+3)}=\frac{4}{a(a-3)}\)

Ответ: \(\frac{4}{a(a-3)}\)

Решение №1879: \(\frac{x+y}{3(x-y)}+\frac{x^{2}}{(x-y)^{2}}=\frac{(x+y)(x-y)+x^{2} \cdot 3}{3(x-y)^{2}}=\frac{x^{2}-y^{2}+3x^{2}}{3(x-y)^{2}}=\frac{4x^{2}-y^{2}}{3(x-y)^{2}}\)

Ответ: \(\frac{4x^{2}-y^{2}}{3(x-y)^{2}}\)

Решение №1883: \(\frac{c-2}{6c+4}-\frac{c-6}{15c+10}=\frac{c-2}{2(3c+2)}-\frac{c-6}{5(3c+2)}=\frac{5(c-2)-2(c-6)}{2 \cdot 5(3c+2)}=\frac{5c-10-2c+12}{10(3c+2)}=\frac{3c+2}{10(3c+2)}=\frac{1}{10}\)

Ответ: \(\frac{1}{10}\)

Решение №1894: \(\frac{3+c}{c^{2}-cd}+\frac{3+d}{d^{2}-cd}=\frac{3+c}{c(c-d)}+\frac{3+d}{d(d-c)}=\frac{3+c}{c(c-d)}-\frac{3+d}{d(c-d)}=\frac{d(3+c)-c(3+d)}{cd(c-d)}=\frac{3d+cd-3c-cd}{cd(c-d)}=\frac{3d-3c}{cd(c-d)}=\frac{-3(c-d)}{cd(c-d)}=-\frac{3}{cd}\)

Ответ: \(-\frac{3}{cd}\)

Решение №1895: \(\frac{3m+n}{9m^{2}-3mn}-\frac{4n}{9m^{2}-n^{2}}=\frac{3m+n}{3m(3m-n)}-\frac{4n}{(3m-n)(3m+n)}=\frac{(3m+n)(3m+n)-4n \cdot 3m}{3m(3m-n)(3m+n)}=\frac{(3m+n)^{2}-12mn}{3m(3m-n)(3m+n)}=\frac{9m^{2}+6mn+n^{2}-12mn}{3m(3m-n)(3m+n)}=\frac{9m^{2}-6mn+n^{2}}{3m(3m-n)(3m+n)}=\frac{(3m-n)^{2}}{3m(3m-n)(3m+n)}=\frac{3m-n}{3m(3m+n)}\)

Ответ: \(\frac{3m-n}{3m(3m+n)}\)

Решение №1901: \(\frac{7n^{2}+mn-8m^{2}}{m^{2}-2mn+n^{2}}-\frac{8m}{n-m}=\frac{7n^{2}+mn-8m^{2}}{(m-n)^{2}}+\frac{8}{m-n}=\frac{7n^{2}+mn-8m^{2}+8m(m-n)}{(m-n)^{2}}=\frac{7n^{2}+mn-8m^{2}+8m^{2}-8nm}{(m-n)^{2}}=\frac{7n^{2}-7mn}{(m-n)^{2}}=\frac{7n(n-m)}{(m-n)^{2}}=\frac{7n(n-m)}{(n-m)^{2}}=\frac{7n}{n-m}\)

Ответ: \(\frac{7n}{n-m}\)