Решение №12026: \(\frac{x}{x+y}+\frac{y}{x-y}=\frac{x(x-y)+y(x+y)}{(x+y)(x-y)}=\frac{-x^{2}-xy+xy+y^{2}}{x^{2}-y^{2}}=\frac{x^{2}+y^{2}}{x^{2}-y^{2}}\)

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

Решение №12029: \(\frac{p+2}{p+1}-\frac{p+6}{p+3}=\frac{(p+2)(p+3)-(p+6)(p+1)}{(p+1)(p+3)}=\frac{p^{2}+3p+2p+6 \cdot (p^{2}+p+6p+6)}{(p+1)(p+3)}=\frac{p^{2}+5p+6-p^{2}-7p-6}{(p+1)(p+3)}=\frac{-2p}{(p+1)(p+3)}\)

Ответ: \(\frac{-2p}{(p+1)(p+3)}\)

Решение №12030: \(\frac{c-d}{2d(c+d)}+\frac{c+d}{2d(c-d)}=\frac{(c-d)(c-d)+(c+d)(c+d)}{2d(c+d)(c-d)}=\frac{(c-d)^{2}+(c+d)^{2}}{2d(c+d)(c-d)}=\frac{c^{2}-2cd+d^{2}+c^{2}+2cd+d^{2}}{2d(c^{2}-d^{2})}=\frac{2c^{2}+2d^{2}}{2d(c^{2}-d^{2})}=\frac{2(c^{2}+d^{2})}{2d(c^{2}-d^{2})}=\frac{c^{2}+d^{2}}{d(c^{2}-d^{2})}\)

Ответ: \(\frac{c^{2}+d^{2}}{d(c^{2}-d^{2})}\)

Решение №12031: \(\frac{x+4y}{5y(x+y)}-\frac{x-y}{5y(x-4y)}=\frac{(x+4y)(x-4y)-(x-y)(x+y)}{5y(x+y)(x-4y)}=\frac{x^{2}-16y^{2}-(x^{2}-y^{2})}{5y(x+y)(x-4y)}=\frac{x^{2}-16-x^{2}+y^{2}}{5y(x+y)(x-4y)}=\frac{-15y^{2}}{5y(x+y)(x-4y)}=\frac{-3y}{(x+y)(x-4y)}\)

Ответ: \(\frac{-3y}{(x+y)(x-4y)}\)

Решение №12033: \(\frac{d-c}{3c(2c+d)}+\frac{2c-d}{3c(c+d)}=\frac{(d-c)(c+d)+(2c-d)(2c+d)}{3c(2c+d)(c+d)}=\frac{d^{2}-c^{2}-(4c^{2}-d^{2})}{3c(2c+d)(c+d)}=\frac{d^{2}-c^{2}-4c^{2}+d^{2}}{3c(2c+d)(c+d)}=\frac{2d^{2}-5c^{2}}{3c(2c+d)(c+d)}\)

Ответ: \(\frac{2d^{2}-5c^{2}}{3c(2c+d)(c+d)}\)

Решение №12035: \(\frac{a-3c}{a-c}+\frac{a^{2}+3c^{2}}{(a-c)(a+c)}=\frac{(a-3c)(a+c)+a^{2}+3c^{2}}{(a-c)(a+c)}=\frac{a^{2}+ac-3ac-3c^{2}+a^{2}+3c^{2}}{(a-c)(a+c)}=\frac{2a^{2}-2ac}{(a-c)(a+c)}=\frac{2a(a-c)}{(a-c)(a+c)}=\frac{2a}{a+c}\)

Ответ: \(\frac{2a}{a+c}\)

Решение №12039: \(\frac{15x-15y}{(5x-3y)(5x+3y)}+\frac{4}{-3y-5x}=\frac{15x-15y}{(5x-3x)(5x+3y)}-\frac{4}{3y+5x}=\frac{15x-15y-4(5x-3y)}{(5x-3y)(5x+3y)}=\frac{15x-15y-20x+12y}{(5x-3y)(5x+3y)}=\frac{-5x-3y}{(5x-3y)(5x+3y)}=\frac{-(5x+3y)}{(5x-3y)(5x+3y)}=-\frac{1}{5x-3y}=\frac{1}{3y-5x}\)

Ответ: \(\frac{1}{3y-5x}\)

Решение №12042: \(\frac{3-x}{(x-1)(x+1)}-\frac{x-2}{x(1-x)}=\frac{3-x}{(x-1)(x+1)}+\frac{x-2}{x(x-1)}=\frac{x(3-x)+(x-2)(x+1)}{x(x-1)(x+1)}=\frac{3x-x^{2}+x^{2}+x-2x-2}{x(x-1)(x+1)}=\frac{2x-2}{x(x-1)(x+1)}=\frac{2(x-1)}{x(x-1)(x+1)}=\frac{2}{x(x+1)}\)

Ответ: \(\frac{2}{x(x+1)}\)

Решение №12047: \(\frac{3m}{(m+5)^{2}}+\frac{2}{m+5}=\frac{3m+2(m+5)}{(m+5)^{2}}=\frac{3m+2m+10}{(m+5)^{2}}=\frac{5m+10}{(m+5)^{2}}=\frac{5(m+2)}{(m+5)^{2}}\)

Ответ: \(\frac{5(m+2)}{(m+5)^{2}}\)

Решение №12051: \(\frac{c}{3d}-\frac{4cd+c^{2}}{3d^{2}+3cd}=\frac{c}{3d}-\frac{4cd+c^{2}}{3d(d+c)}=\frac{c(d+c)-4cd-c^{2}}{3d(d+c)}=\frac{cd+c^{2}-4cd-c^{2}}{3d(d+c)}=\frac{cd-4cd}{3d(d+c)}=\frac{-3cd}{3d(d+c)}=\frac{-3cd}{3d(d+c)}=-\frac{c}{d+c}\)

Ответ: \(-\frac{c}{d+c}\)

Решение №12054: \(\frac{b+2a}{a^{2}+ab}-\frac{a+2b}{b^{2}+ab}=\frac{b+2a}{a(a+b)}-\frac{a+2b}{b(b+a)}=\frac{b(b+2a)-a(a+2b)}{ab(a+b)}=\frac{b^{2}+2ab-a^{2}-2ab}{ab(a+b)}=\frac{b^{2}-a^{2}}{ab(a+b)}=\frac{(b-a)(b+a)}{ab(a+b)}=\frac{b-a}{ab}\)

Ответ: \(\frac{b-a}{ab}\)

Решение №12055: \(\frac{d+3}{cd+d^{2}}-\frac{c-3}{cd+c^{2}}=\frac{d+3}{d(c+d)}-\frac{c-3}{c(d+c)}=\frac{c(d+3)-d(c-3)}{cd(d+c)}=\frac{cd+3c-cd+3d}{cd(d+c)}=\frac{3c+3d}{cd(d+c)}=\frac{3(c+d)}{cd(d+c)}=\frac{3}{cd}\)

Ответ: \(\frac{3}{cd}\)

Решение №12058: \(\frac{10x}{16-x^{2}}+\frac{5}{x-4}=\frac{10x}{(4-x)(4+x)}-\frac{5}{x-4}=\frac{10x+5(x+4)}{(4-x)(4+x)}=\frac{10x-5x-20}{(4-x)(4+x)}=\frac{5x-20}{(4-x)(4+x)}=\frac{-5(4-x)}{(4-x)(4+x)}=-\frac{5}{4+x}\)

Ответ: \(-\frac{5}{4+x}\)

Решение №12065: \(\frac{6x^{2}-15x+25}{4x^{2}-25}+\frac{x}{5-2x}=\frac{6x^{2}-15x+25}{(2x-5)(2x+5)}-\frac{x}{2x-5}=\frac{6x^{2}-15x+25-x(2x+5)}{(2x-5)(2x+5)}=\frac{6x^{2}-15x+25-2x^{2}-5x}{(2x5)(2x+5)}=\frac{4x^{2}-20x+25}{(2x-5)(2x+5)}=\frac{(2x-5)^{2}}{(2x-5)(2x+5)}=\frac{2x-5}{2x+5}\)

Ответ: \(\frac{2x-5}{2x+5}\)

Решение №12071: \(\frac{a+b}{a^{2}-ab}-\frac{a}{(b-a)^{2}}=\frac{a+b}{a(a-b)}-\frac{a}{(a-b)^{2}}=\frac{(a+b)(a-b)-a^{2}}{a(a-b)^{2}}=\frac{a^{2}-b^{2}-a^{2}}{a(a-b)^{2}}=-\frac{b^{2}}{a(a-b)^{2}}\)

Ответ: \(-\frac{b^{2}}{a(a-b)^{2}}\)

Решение №12073: \(\frac{2x^{2}+1}{x^{3}-1}-\frac{x}{x^{2}+x+1}=\frac{2x^{2}+1}{(x-1)(x^{2}+x+1)}-\frac{x}{x^{2}+x+1}=\frac{2x^{2}+1-x(x-1)}{(x-1)(x^{2}+x+1)}=\frac{2x^{2}+1-x^{2}+x}{(x-1)(x^{2}+x+1)}=\frac{x^{2}+x+1}{(x-1)(x^{2}+x+1)}=\frac{1}{x-1}\)

Ответ: \(\frac{1}{x-1}\)

Решение №12075: \(\frac{6c^{2}+48}{c^{3}+64}-\frac{3c}{c^{2}-4c+16}=\frac{6c^{2}+48}{(c+4)(c^{2}-4c+16)}-\frac{3c}{c^{2}-4c+16}=\frac{6c^{2}+48-3c(c+4)}{(c+4)(c^{2}-4c+16)}=\frac{6c^{2}+48-3c^{2}-12c}{(c+4)(c^{2}-4c+16)}=\frac{3c^{2}-12c+48}{(c+4)(c^{2}-4c+16)}=\frac{3(c^{2}-4c+16)}{(c+4)(c^{2}-4c+16)}=\frac{3}{c+4}\)

Ответ: \(\frac{3}{c+4}\)

Решение №12077: \(c^{2}-cd+d^{2}-\frac{c^{3}-d^{3}}{c+d}=c^{2}-cd+d^{2}-\frac{(c-d)(c^{2}+cd+d^{2})}{(c+d)}=\frac{(c+d)(c^{2}-cd+d^{2})-c^{3}+d^{3}}{(c+d)}=\frac{c^{3}+d^{3}-c^{3}+d^{3}}{c+d}=\frac{2d^{3}}{c+d}\)

Ответ: \(\frac{2d^{3}}{c+d}\)

Решение №12084: \(\frac{c}{c^{2}+3c+9}-\frac{1}{c-3}+\frac{27}{c^{3}-27}=\frac{c}{c^{2}+3c+9}-\frac{1}{c-3}+\frac{27}{(c-3)(c^{2}+3c+9)}=\frac{c(c-3)-(c^{2}+3c+9)+27}{(c-3)(c^{2}+3c+9)}=\frac{c^{2}-3c-c^{2}-3c-9+27}{c^{3}-27}=\frac{18}{c^{3}-27}\)

Ответ: \(\frac{18}{c^{3}-27}\)

Решение №12085: \(1-\frac{2d-1}{4d^{2}-2d+1}-\frac{2d}{2d+1}=\frac{(2d+1)(4d^{2}-2d+1)-(2d-1)(2d+1)}{(2d+1)(4d^{2}-2d+1)}-\frac{2d(4d^{2}-2d+1)}{(2d+1)(4d^{2}-2d+1)}=\frac{8d^{3+1-(4d^{2}-1)-(8d^{3}-4d^{2}+2d)}{8d^{3}+1}=\frac{8d^{3}+1-4d^{2}+1-8d^{3}+4d^{2}-2d}{8d^{3}+1}=\frac{2-2d}{8d^{3}+1}\)

Ответ: \(\frac{2-2d}{8d^{3}+1}\)

Решение №12092: \(\frac{1}{(2m-5n)^{2}}-\frac{2}{25n^{2}-4m^{2}}+\frac{1}{(5n+2m)^{2}}=\frac{1}{(2m-5n)^{2}-\frac{2}{(5n-2m)(5n+2m)}+\frac{1}{(5n+2m)^{2}}=\frac{1}{(5n-2m)^{2}}-\frac{2}{(5n-2m)(5n+2m)}+\frac{1}{(5n+2m)^{2}}=\frac{(5n+2m)^{2}}{(5n-2m)^{2}(5n+2m)^{2}}-\frac{2(5n-2m)(5n+2m}{(5n-2m)^{2}(5n+2m)^{2}}+\frac{(5n-2m)^{2}}{(5n+2m)^{2}(5n-2m)^{2}}=\frac{25n^{2}+20mn+4m^{2}}{(5n-2m)^{2}(5n+2m)^{2}}-\frac{2(25n^{2}-4m^{2})}{(5n-2m)^{2}(5n+2m)^{2}}+\frac{25n^{2}-20mn+4m^{2}}{(5n-2m)^{2}(5n+2m)^{2}}=\frac{25n^{2}+20mn+4m^{2}-50n^{2}+8m^{2}+25n^{2}-20mn+4m}{(5n-2m)^{2}(5n+2m)^{2}}=\frac{16m^{2}}{(5n-2m)^{2}(5m+2m)^{2}}=\frac{16m^{2}}{(25m^{2}-4m^{2})^{2}}=\frac{16m^{2}}{(4m^{2}-25m^{2})^{2}}\)

Ответ: \(\frac{16m^{2}}{(4m^{2}-25m^{2})^{2}}\)

Решение №12095: \(\frac{1}{2z^{2}+5z}-\frac{2}{25-10z}-\frac{4}{4z^{2}-25}=\frac{1}{5z}=\frac{1}{z(2z+5)}-\frac{2}{5(5-2z)}-\frac{4}{(2z-5)(2z+5)}=\frac{(5-2z) \cdot 5}{5z(5+2z)(5-2z)}-\frac{2z(5+2z)}{5(5-2z)(5+2z)z}+\frac{4 \cdot 5 \cdot z}{5z(5-2z)(5+2z)}=\frac{25-10z-10z-4z^{2}+20z}{5z(5+2z)(5-2z)}=\frac{25-4z^{2}}{5z(5+2z)(5-2z}=\frac{25-4z^{2}}{5z(25-4z^{2})}=\frac{1}{5z}\)

Ответ: NaN

Решение №12099: \(\frac{a+3b}{\frac{a}{a+b}+\frac{b}{a-b}}=\frac{a+3b}{\frac{a(a-b)+b(a+b)}{(a+b)(a-b)}}=\frac{(a+3b)(a^{2}-b^{2})}{a^{2}-ab+ab+b^{2}}=\frac{(a+3b)(a^{2}-b^{2})}{a^{2}+b^{2}}; a+b \neq 0, a \neq -b; a-b \neq 0, a \neq b\)

Ответ: \(a \neq b\)

Решение №12107: \((\frac{b}{c}+\frac{c}{b})^{2}+(\frac{c}{a}+\frac{a}{c})^{2}+(\frac{a}{b}+\frac{b}{a})^{2}-(\frac{b}{c}+\frac{c}{b})(\frac{c}{a}+\frac{a}{c})(\frac{a}{b}+\frac{b}{a})=(\frac{b^{2}+c^{2}}{bc})^{2}+(\frac{c^{2}+a^{2}}{ac})^{2}+(\frac{a^{2}+b^{2}}{ab})^{2}-(\frac{b^{2}+b^{2}}{ab})^{2} \cdot (\frac{c^{2}+a^{2}}{ac})(\frac{a^{2}+b^{2}}{ab})=\frac{(b^{2}+c^{2})^{2}}{b^{2}c^{2}}+\frac{(c^{2}+a^{2})^{2}}{a^{2}c^{2}}+\frac{(a^{2}+b^{2})^{2}}{a^{2}b^{2}}-\frac{(b^{2}+c^{2})(c^{2}+a^{2})(a^{2}+b^{2})}{a^{2}b^{2}c^{2}}=\frac{b^{4}+2b^{2}+c^{4}}{b^{2}c^{2}}+\frac{c^{4}+2c^{2}a^{2}+a^{4}}{a^{2}c^{2}}+\frac{a^{4}+2a^{2}b^{2}+b^{4}}{a^{2}b^{2}}-\frac{(b^{2}c^{2}+a^{2}b^{2}+c^{4}+a^{2}c^{2})(a^{2+b^{2})}{a^{2}b^{2}c^{2}}=\frac{a^{2}(b^{4}+2b^{2}c^{2}+c^{4})+b^{2}(c^{4}+2c^{2}a^{2}+a^{4})+c^{2}(a^{4}+2a^{2}b^{2}+b^{4}}{a^{2}b^{2}c^{2}}-\frac{a^{2}b^{2}c^{2}+a^{4}b^{2}+a^{2}c^{4}+a^{4}c^{2}+b^{4}c^{2}+a^{2}b^{4}+b^{2}c^{4}+a^{2}b^{2}c^{2}}{a^{2}b^{2}c^{2}}=\frac{6a^{2}b^{2}c^{2}-2a^{2}b^{2}c^{2}}{a^{2}b^{2}c^{2}}=\frac{4a^{2}b^{2}c^{2}}{a^{2}b^{2}c^{2}}=4\)

Ответ: \(4\)

Решение №12118: \(\frac{4}{3y^{3}} \cdot \frac{y^{8}}{18}=frac{4 \cdot y^{8}}{3y^{3} \cdot 18}=\frac{2 \cdot y^{5}}{3 \cdot 9}=\frac{2y^{5}}{27}\)

Ответ: \(\frac{2y^{5}}{27}\)

Решение №12121: \(\frac{a^{3}b}{c} \cdot \frac{c^{2}}{a^{4}b^{2}}=frac{a^{3}bc^{2}}{ca^{4}b^{2}}=\frac{c}{ab}\)

Ответ: \(\frac{c}{ab}\)

Решение №12122: \(\frac{3m^{2}n}{c}:\frac{m^{2}n}{3c}=frac{3m^{2}n}{c} \cdot \frac{3c}{m^{2}n}=\frac{3m^{2}n \cdot 3c}{c \cdot n^{2}n}=9\)

Ответ: \(9\)

Решение №12123: \(\frac{m^{3}}{cd}:\frac{m^{2}n}{cd}=frac{m^{3}}{cd} \cdot \frac{cd}{m^{2}n}=\frac{m^{3}cd}{cdm^{2}n}=\frac{m}{n}\)

Ответ: \(\frac{m}{n}\)

Решение №12130: \(\frac{a+b}{8}:\frac{a+b}{8x}=frac{a+b}{8} \cdot \frac{8x}{a+b}=\frac{(a+b)8x}{8(a+b)}=x\)

Ответ: \(x\)

Решение №12131: \(\frac{2m-3n}{7} \cdot \frac{7s}{2m-3n}=frac{(2m-3n) \cdot 7s}{7 \cdot (2m-3n)}=s\)

Ответ: \(s\)

Решение №12137: \(\frac{c+d}{c-d} \cdot \frac{c-d}{c(c+d)}=frac{(c+d)(c-d)}{(c-d) \cdot c \cdot (c+d)}=\frac{1}{c}\)

Ответ: \(\frac{1}{c}\)

Решение №12138: \(\frac{a-b}{c+d}:\frac{3(a-b)}{a(c+d)}=frac{a-b}{c+d} \cdot \frac{a(c+d)}{3(a-b)}=\frac{(a-b) \cdot a \cdot (c+d)}{(c+d) \cdot 3 \cdot (c-b)}=\frac{a}{3}\)

Ответ: \(\frac{a}{3}\)

Решение №12142: \(\frac{2x+y}{x-y}:(2x+y)^{2}=frac{2x+y}{x-y} \cdot \frac{1}{(2x+y)^{2}}=\frac{(2x+y)}{(x-y)(2x+y)^{2}}=\frac{1}{(x-y)(2x+y)}\)

Ответ: \(\frac{1}{(x-y)(2x+y)}\)

Решение №12144: \(\frac{mx+my}{ab^{2}} \cdot \frac{a^{2}b}{4x+4y}=frac{m(x+y) \cdot a^{2}b}{ab^{2} \cdot 4(x+y)}=\frac{ma}{4b}\)

Ответ: \(\frac{ma}{4b}\)

Решение №12151: \(\frac{x^{2}-y^{2}}{3xy} \cdot \frac{3y}{x-y}=frac{(x-y)(x+y) \cdot 3y}{3xy(x-y)}=\frac{x+y}{x}\)

Ответ: \(\frac{x+y}{x}\)

Решение №12161: \(\frac{12n}{x^{3}-27} \cdot \frac{x^{2}+3x+9}{6n}=frac{12n \cdot (x^{2}+3x+9)}{(x-3)(x^{2}+3x+9)6n}=\frac{2}{x-3}\)

Ответ: \(\frac{2}{x-3}\)

Решение №12163: \(\frac{x^{2}-10x+25}{3x+12}:\frac{2x-10}{x^{2}-16}=frac{(x-5)^{2}}{3(x+4)} \cdot \frac{(x-4)(x+4)}{2(x-5)}=\frac{(x-5)^{2} \cdot (x-4)(x+4)}{3(x+4) \cdot 2(x-5)}=\frac{(x-5)(x-4)}{6}\)

Ответ: \(\frac{(x-5)(x-4)}{6}\)

Решение №12164: \(\frac{1-a}{4a+8b} \cdot \frac{a^{2}+4ab+4b^{2}}{3-3a}=frac{(1-a) \cdot (a+2b)^{2}}{4(a+2b) \cdot 3(1-a)}=\frac{a+2b}{12}\)

Ответ: \(\frac{a+2b}{12}\)

Решение №12169: \((\frac{8z}{3})^{2}=frac{(8x)^{2}}{9x^{2}}=\frac{64x^{2}}{91}\)

Ответ: \(\frac{64x^{2}}{91}\)

Решение №12170: \((\frac{t}{4b})^{4}=\frac{t^{4}}{(4b)^{4}}=\frac{t^{4}}{256b^{4}}\)

Ответ: \(\frac{t^{4}}{256b^{4}}\)

Решение №12171: \((-\frac{2x}{3y})^{5}=-frac{(2x)^{5}}{(3y)^{5}}=-\frac{32x^{5}}{243y^{5}}

Ответ: \(-\frac{32x^{5}}{243y^{5}}

Решение №12174: \((-\frac{3m}{4n})^{4}=frac{(3m)^{4}}{(4n)^{4}}=\frac{81m^{4}}{256n^{4}}\)

Ответ: \(\frac{81m^{4}}{256n^{4}}\)

Решение №12176: \((-\frac{3n^{6}k^{3}}{10p^{4}})^{3}=-frac{(3n^{6}k^{3})^{3}}{(10p^{4})^{3}}=-\frac{3^{3}(n^{6})^{3}(k^{3})^{3}}{10^{3}(p^{4})^{3}}=-\frac{27n^18}k^{4}}{1000p^{12}}\)

Ответ: \(-\frac{27n^18}k^{4}}{1000p^{12}}\)

Решение №12177: \((\frac{5a^{4}c^{3}}{2k^{3}})^{3}=\frac{(5a^{4}c^{3})^{3}}{(2k^{3})^{3}}=\frac{5^{3}(a^{4})^{3}(c^{3})^{3}}{2^{3}(k^{3})^{3}}=\frac{125a^{12}c^{9}}{8k^{9}}\)

Ответ: \(\frac{125a^{12}c^{9}}{8k^{9}}\)

Решение №12178: \((-\frac{5x^{6}y^{3}}{z^{8}})^{4}=\frac{(5x^{6}y^{3})^{4}}{(z^{8})^{4}}=\frac{5^{4}(x^{6})^{4}(y^{3})^{4}}{z^{32}}=\frac{625x^{24}y^{12}}{z^{32}}\)

Ответ: \(\frac{625x^{24}y^{12}}{z^{32}}\)