\(1)\ \frac{x^2+14x+49}{x+6}:\left(\frac{13}{x+6}-x+6\right)=\frac{7+x}{7-x}\)
\(1)\ \frac{13}{x+6}-x+6={\frac{13}{x+6}}^{\backslash1}-{\frac{x}{1}}^{\backslash{x\ +\ 6}}+{\frac{6}{1}}^{\backslash{x\ +\ 6}}=\frac{13-(x^2+6x)+6x+36}{x+6}=\frac{13-x^2-6x+6x+36}{x+6}=\frac{{-}x^2+49}{x+6}=\frac{ {-}(x^2-49)}{x+6}={-}\frac{x^2-49}{x+6}\)
\(2)\ \frac{x^2+14x+49}{x+6}:\left({-}\frac{x^2-49}{x+6}\right)=\frac{ (x+7)^2}{x+6}\cdot\left({-}\frac{x+6}{ (x-7)(x+7)}\right)={-}\frac{x+7}{x-7}=\frac{7+x}{7-x}\)
\(2)\ \left(c-\frac{2c-9}{c+8}\right):\frac{c^2+3c}{c^2-64}+\frac{24}{c}=c-5\)
\(1)\ c-\frac{2c-9}{c+8}={\frac{c}{1}}^{\backslash{c\ +\ 8}}-{\frac{2c-9}{c+8}}^{\backslash1}=\frac{c^2+8c-(2c-9)}{c+8}=\frac{c^2+8c-2c+9}{c+8}=\frac{c^2+6c+9}{c+8}=\frac{ (c+3)^2}{c+8}\)
\(2)\ \frac{ (c+3)^2}{c+8}:\frac{c^2+3c}{c^2-64}=\frac{ (c+3)^2}{c+8}\cdot\frac{ (c-8)(c+8)}{ c(c+3)}=\frac{ (c+3)(c-8)}{c}=\frac{c^2-8c+3c-24}{c}=\frac{c^2-5c-24}{c}\)
\(3)\ {\frac{c^2-5c-24}{c}}^{\backslash1}+{\frac{24}{c}}^{\backslash1}=\frac{c^2-5c-24+24}{c}=\frac{c^2-5c}{c}=\frac{ c(c-5)}{c}=c-5\)
\(3)\ \left(\frac{36}{x^2-9}-\frac{x-3}{x+3}-\frac{3+x}{3-x}\right):\frac{6}{3-x}={-}2\)
\(1)\ \frac{36}{x^2-9}-\frac{x-3}{x+3}-\frac{3+x}{3-x}=\frac{36}{x^2-9}-\frac{x-3}{x+3}-\frac{3+x}{ {-}(x-3)}={\frac{36}{ (x-3)(x+3)}}^{\backslash1}-{\frac{x-3}{x+3}}^{\backslash{x\ -\ 3}}+{\frac{x+3}{x-3}}^{\backslash{x\ +\ 3}}=\frac{36-(x-3)^2+(x+3)^2}{ (x-3)(x+3)}=\frac{36-(x^2-6x+9)+x^2+6x+9}{ (x-3)(x+3)}=\frac{36-x^2+6x-9+x^2+6x+9}{ (x-3)(x+3)}=\frac{12x+36}{ (x-3)(x+3)}=\frac{ 12(x+3)}{ (x-3)(x+3)}=\frac{12}{x-3}\)
\(2)\ \frac{12}{x-3}:\frac{6}{3-x}=\frac{12}{x-3}:\frac{6}{ {-}(x-3)}=\frac{12}{x-3}\cdot\left({-}\frac{x-3}{6}\right)={-}2\)
\(4)\ \left(\frac{2y-1}{y^2+2y+4}+\frac{9y+6}{y^3-8}+\frac{1}{y-2}\right)\cdot\frac{y^2-4}{18}=\frac{y+2}{6}\)
\(1)\ \frac{2y-1}{y^2+2y+4}+\frac{9y+6}{y^3-8}+\frac{1}{y-2}={\frac{2y-1}{y^2+2y+4}}^{\backslash{y\ -\ 2}}+{\frac{9y+6}{ (y-2)(y^2+2y+4)}}^{\backslash1}+{\frac{1}{y-2}}^{\backslash{y^2\ +\ 2y\ +\ 4}}=\frac{2y^2-4y-y+2+9y+6+y^2+2y+4}{ (y-2)(y^2+2y+4)}=\frac{3y^2+6y+12}{ (y-2)(y^2+2y+4)}=\frac{ 3(y^2+2y+4)}{ (y-2)(y^2+2y+4)}=\frac{3}{y-2}\)
\(2)\ \frac{3}{y-2}\cdot\frac{y^2-4}{18}=\frac{3}{y-2}\cdot\frac{ (y-2)(y+2)}{18}=\frac{y+2}{6}\)
