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: \(\frac{7n}{n-m}\)

Решение №1910: \(\frac{m^{3}+n^{3}}{m-n}-m^{2}-mn-n^{2}=\frac{m^{3}+n^{3}-(m-n)(m^{2}+mn+n^{2})}{m-n}=\frac{m^{3}+n^{3}-(m^{3}-n^{3})}{m-n}=\frac{m^{3}+n^{3}-m^{3}+n^{3}}{m-n}=\frac{2n^{3}}{m-n}\)

Ответ: \(\frac{2n^{3}}{m-n}\)

Решение №1911: \(\frac{x^{3}+y^{3}}{x^{2}+xy+y^{2}}+x-y=\frac{x^{3}+y^{3}(x-y)(x^{2}+xy+y^{2})}{x^{2}+xy+y^{2}}=\frac{x^{3}+y^{3}+x^{3}-y^{3}}{x^{2}+xy+y^{2}}=\frac{2x^{3}}{x^{2}+xy+y^{2}}\)

Ответ: \(\frac{2x^{3}}{x^{2}+xy+y^{2}}\)

Решение №1914: \(1-\frac{1}{a^{3}+1}-\frac{a}{a+1}=1-\frac{1}{(a+1)(a^{2}-a+1)}-\frac{a}{a+1}=\frac{a^{3}+1-1-a(a^{2}-a+1)}{a^{3}+1}=\frac{a^{3}-a^{3}+a^{2}-a}{a^{3}+1}=\frac{a^{2}-a}{a^{3}+1}\)

Ответ: \(\frac{a^{2}-a}{a^{3}+1}\)

Решение №1917: \(\frac{1}{b+2}-\frac{b}{b^{2}-2b+4}-\frac{12}{b^{3}+8}=\frac{b^{2}-2b+4}{b^{3}+8}-\frac{b(b+2)}{b^{3}+8}-\frac{12}{b^{3}+8}=\frac{b^{2}-2b+4-b^{2}-2b-12}{b^{3}+8}=\frac{-4b-8}{b^{3}+8}=\frac{-4(b+2)}{(b+2)(b^{2}-2b+4)}=-\frac{4}{b^{2}-2b+4}=\frac{4}{2b-b^{2}-4}\)

Ответ: \(\frac{4}{2b-b^{2}-4}\)

Решение №1925: \(\frac{x+2y}{x^{2}+2xy+y^{2}}-\frac{x-2y}{x^{2}-y^{2}}+\frac{2y^{2}}{(x+y)(x^{2}-y^{2})}=\frac{2y}{x^{2}-y^{2}}=\frac{x+2y}{(x+y)^{2}}-\frac{x-2y}{(x-y)(x+y)}+\frac{2y^{2}}{(x+y)(x-y)(x+y)}=\frac{(x+2y)(x-y)}{(x+y)^{2}(x-y)}-\frac{(x-2y)(x+y)}{(x+y)^{2}(x-y)}+\frac{2y^{2}}{(x+y)^{2}(x-y)}=\frac{x^{2}-xy+2xy-2y^{2}-(x^{2}+xy-2xy-2y^{2})+2y^{2}}{(x+y)^{2}(x-y)}=\frac{2xy+2y^{2}}{(x+y)^{2}(x-y)}=\frac{2y(x+y)}{(x+y)^{2}(x-y)}=\frac{2y}{(x+y)(x-y)}=\frac{2y}{x^{2}-y^{2}}\)

Ответ: NaN

Решение №1931: \(\frac{x-2y}{x-\frac{x^{2}+y^{2}}{x-y}}=\frac{x-2y}{\frac{x(x-y)-x^{2}-y^{2}}{x-y}}=\frac{(x-2y)(x-y)}{x^{2}-xy-x^{2}-y^{2}}=\frac{(x-2y)(x-y)}{-xy-y}=-\frac{(x-2y)(x-y)}{y(x+y)}; x-y \neq 0, x \neq y, y \neq 0, x+y \neq 0, x \neq -y\)

Ответ: \(x \neq -y\)

Решение №1932: \(\frac{8}{(x^{2}+3}+\frac{2}{x^{2}+3}+\frac{1}{x+1}=\frac{8}{(x^{2}+3)(x=1)(x+1)}+\frac{2(x^{2}-1)}{(x^{2}+3)(x^{2}-1)}+\frac{(x^{2}+3)(x-1)}{(x^{2}+3)(x^{2}-1)}=\frac{8+2x^{2}-2+x^{3}-x^{2}+3x-3}{(x^{2}+3)(x^{2}-1)}=\frac{x^{3}+x^{2}+3x+3}{(x^{2}+3)(x^{2}-1)}=\frac{x^{2}(x+1)+3(x+1)}{(x^{2}+3)(x^{2}-1)}=\frac{(x^{2}+3)(x+1)}{(x^{2}+3)(x-1)(x+1)}=\frac{1}{x-1}; x=0,992; \frac{1}{0,992-1}=\frac{1}{-0,008}=-\frac{1000}{8}=-125\)

Ответ: \(-125\)

Решение №1941: \(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+ … +\frac{1}{99 \cdot 100}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+…+\frac{1}{99}-\frac{1}{100}=1-\frac{1}{100}-\frac{100}{100}-\frac{1}{100}=\frac{99}{100}=0,99\)

Ответ: \(0,99\)

Решение №1943: \(\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 6}+ … +\frac{1}{99 \cdot 101}=\frac{1}{2}(\frac{1}{1}-\frac{1}{3})+\frac{1}{2}(\frac{1}{3}-\frac{1}{5})+\frac{1}{2}(\frac{1}{5}-\frac{1}{7})+…+\frac{1}{2}(\frac{1}{99}-\frac{1}{100})=\frac{1}{2}(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101})=\frac{1}{2}(1-\frac{1}{101})=\frac{1}{2} \cdot (\frac{101}{101}-\frac{1}{101})=\frac{1}{2} \cdot \frac{100}{101}=\frac{50}{101}\)

Ответ: \(\frac{50}{101}\)

Решение №1947: \(\frac{5c}{2d}:(-\frac{15c}{d})=frac{5c}{2d} \cdot (-\frac{d}{15c})=-\frac{5c \cdot d}{2d \cdot 15c}=-\frac{5}{30}=-\frac{1}{6}\)

Ответ: \(-\frac{1}{6}\)

Решение №1948: \(\frac{12x^{5}}{55}:\frac{6x^{2}}{5}=frac{12x^{5}}{55} \cdot \frac{5}{6x^{2}}=\frac{12x^{5} \cdot 5}{55 \cdot 6x^{2}}=\frac{2x^{3}}{11}\)

Ответ: \(\frac{2x^{3}}{11}\)

Решение №1950: \(\frac{16}{5d^{3}}:\frac{12}{d^{4}}=frac{16}{5d^{3}} \cdot \frac{d^{4}}{12}=\frac{16 \cdot d^{4}}{5d^{3} \cdot 12}=\frac{4d}{5 \cdot 3}=\frac{4d}{15}\)

Ответ: \(\frac{4d}{15}\)

Решение №1951: \(\frac{3c^{12}}{49} \cdot \frac{7}{6c^{15}}=frac{36c^{12} \cdot 7}{49 \cdot 6c^{15}}=\frac{6}{7 \cdot c^{3}}=\frac{6}{7c^{3}}\)

Ответ: \(\frac{6}{7c^{3}}\)

Решение №1955: \(\frac{5c^{2}x}{a} \cdot \frac{15a}{c^{3}x}=frac{5c^{2}x \cdot 15a}{a \cdot c^{3}x}=\frac{75}{c}\)

Ответ: \(\frac{75}{c}\)

Решение №1957: \(9xy:\frac{3x^{2}y}{ab}=9xy \cdot frac{ab}{3x^{2}y}=\frac{9xy \cdot ab}{3x^{2}y}=\frac{3ab}{x}\)

Ответ: \(\frac{3ab}{x}\)

Решение №1958: \(\frac{m}{17a^{2}d^{2}} \cdot 34a^{2}d^{8}=frac{m \cdot 34a^{2}d^{8}}{17a^{2}d^{2}}=2md^{6}\)

Ответ: \(2md^{6}\)

Решение №1959: \(\frac{4x^{3}y^{3}}{a}:36x^{3}y^{4}=frac{4x^{3}y^{4} \cdot 1}{a \cdot 36x^{3}y^{4}}=\frac{1}{9a}\)

Ответ: \(\frac{1}{9a}\)

Решение №1964: \(\frac{16u-13u}{21}:\frac{13u-16u}{p}=-frac{13v-16u}{p}=-\frac{13v-16u}{21} \cdot \frac{p}{13v-16u}=-\frac{(13v-16u) \cdot p}{21 \cdot (13v-16u)}=-\frac{p}{21}\)

Ответ: \(-\frac{p}{21}\)

Решение №1967: \(\frac{64r-15s}{9c^{2}} \cdot \frac{18c}{15s-64r}=-frac{(15s-64r) \cdot 18c}{9c^{2} \cdot (15s-64r)}=-\frac{2}{3}\)

Ответ: \(-\frac{2}{3}\)

Решение №1971: \(\frac{a+b}{2b(a-b)}:\frac{a+b}{2b^{2}(a-b)}=frac{a+b}{2b(a-b)} \cdot \frac{2b^{2}(a-b)}{(a+b)}=\frac{(a+b) \cdot 2b^{2} \cdot (a-b)}{2b \cdot (a-b)(a+b)}=b\)

Ответ: \(b\)