Упражнение 108

\(1)\ \frac{3}{x+3}+\frac{x+4}{x^2-9}={\frac{3}{x+3}}^{\backslash{x\ -\ 3}}+{\frac{x+4}{ (x-3)(x+3)}}^{\backslash1}=\frac{ 3(x-3)+x+4}{ (x-3)(x+3)}=\frac{3x-9+x+4}{ (x-3)(x+3)}=\frac{4x-5}{ (x-3)(x+3)}=\frac{4x-5}{x^2-9};\)

\(2)\ \frac{a^2}{a^2-64}-\frac{a}{a-8}={\frac{a^2}{ (a-8)(a+8)}}^{\backslash1}-{\frac{a}{a-8}}^{\backslash{a\ +\ 8}}=\frac{a^2-a(a+8)}{ (a-8)(a+8)}=\frac{a^2-a^2-8a}{ (a-8)(a+8)}=\frac{{-}8a}{ (a-8)(a+8)}={-}\frac{8a}{a^2-64};\)

\(3)\ \frac{6b}{9b^2-4}-\frac{1}{3b-2}={\frac{6b}{ (3b-2)(3b+2)}}^{\backslash1}-{\frac{1}{3b-2}}^{\backslash{3b\ +\ 2}}=\frac{6b-(3b+2)}{ (3b-2)(3b+2)}=\frac{6b-3b-2}{ (3b-2)(3b+2)}=\frac{3b-2}{ (3b-2)(3b+2)}=\frac{1}{3b+2};\)

\(4)\ \frac{3a+b}{a^2-b^2}+\frac{1}{a+b}={\frac{3a+b}{ (a-b)(a+b)}}^{\backslash1}+{\frac{1}{a+b}}^{\backslash{a\ -\ b}}=\frac{3a+b+a-b}{ (a-b)(a+b)}=\frac{4a}{ (a-b)(a+b)}=\frac{4a}{a^2-b^2};\)

\(5)\ \frac{m}{m+5}-\frac{m^2}{m^2+10m+25}={\frac{m}{m+5}}^{\backslash{m\ +\ 5}}-{\frac{m^2}{ (m+5)^2}}^{\backslash1}=\frac{ m(m+5)-m^2}{ (m+5)^2}=\frac{m^2+5m-m^2}{ (m+5)^2}=\frac{5m}{ (m+5)^2}=\frac{5m}{m^2+10m+25};\)

\(6)\ \frac{b}{a+b}-\frac{b^2}{a^2+b^2+2ab}={\frac{b}{a+b}}^{\backslash{a\ +\ b}}-{\frac{b^2}{ (a+b)^2}}^{\backslash1}=\frac{ b(a+b)-b^2}{ (a+b)^2}=\frac{ab+b^2-b^2}{ (a+b)^2}=\frac{ab}{ (a+b)^2}=\frac{ab}{a^2+2ab+b^2}.\)