$1) \frac{a{b}^{{\displaystyle \frac{3}{2}}}—b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}.$

Воспользуемся свойствами степеней

${\left(a^n\right)}^m=a^{n·m}, a^{n+m}=a^n·a^m$

и формулой сокращенного умножения

$a^2—b^2=\left(a—b\right)\left(a+b\right).$ $\frac{a{b}^{{\displaystyle \frac{3}{2}}}—b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=\frac{a{b}^{{\displaystyle 1+\frac{1}{2}}}—b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=\frac{a{b}^1{b}^{{\displaystyle \frac{1}{2}}}—b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=\frac{b^{{\displaystyle \frac{1}{2}}}\left(ab—1\right)}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=\frac{b^{{\displaystyle \frac{1}{2}}}\left({\left(ab\right)}^{2·{\displaystyle \frac{1}{2}}}—1\right)}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=\frac{b^{{\displaystyle \frac{1}{2}}}\left({\left({\left(ab\right)}^{{\displaystyle \frac{1}{2}}}\right)}^2—1^2\right)}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=\frac{b^{{\displaystyle \frac{1}{2}}}\left(a^{\frac{1}{2}}{b}^{\frac{1}{2}}—1\right)\left(a^{\frac{1}{2}}{b}^{\frac{1}{2}}+1\right)}{a^{{\displaystyle \frac{1}{2}}}{b}^{{\displaystyle \frac{1}{2}}}—1}=b^{\frac{1}{2}}\left(a^{\frac{1}{2}}{b}^{\frac{1}{2}}+1\right)=a^{\frac{1}{2}}{b}^{\frac{1}{2}}{b}^{\frac{1}{2}}+b^{\frac{1}{2}}=a^{\frac{1}{2}}{b}^{\frac{1}{2}+\frac{1}{2}}+b^{\frac{1}{2}}=a^{\frac{1}{2}}{b}^1+b^{\frac{1}{2}}=a^{\frac{1}{2}}b+b^{\frac{1}{2}}.$ $2) \frac{b}{a—b}+\frac{b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}}.$

Воспользуемся свойствами степеней

${\left(a^n\right)}^m=a^{n·m}, a^{n+m}=a^n·a^m$

и формулой сокращенного умножения

$a^2—b^2=\left(a—b\right)\left(a+b\right).$ $\frac{b}{a—b}+\frac{b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}}=\frac{b}{a^{2·{\displaystyle \frac{1}{2}}}—b^{2·{\displaystyle \frac{1}{2}}}}+\frac{b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}}=\frac{b}{{\left(a^{{\displaystyle \frac{1}{2}}}\right)}^2—{\left(b^{{\displaystyle \frac{1}{2}}}\right)}^2}+\frac{b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}}=\frac{b}{\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)\left(a^{\frac{1}{2}}—b^{\frac{1}{2}}\right)}+\frac{b^{{\displaystyle \frac{1}{2}}}}{a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}}=\frac{b}{\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)\left(a^{\frac{1}{2}}—b^{\frac{1}{2}}\right)}+\frac{b^{{\displaystyle \frac{1}{2}}}\left(a^{\frac{1}{2}}—b^{\frac{1}{2}}\right)}{\left(a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}\right)\left(a^{\frac{1}{2}}—b^{\frac{1}{2}}\right)}=\frac{b+b^{{\displaystyle \frac{1}{2}}}\left(a^{\frac{1}{2}}—b^{\frac{1}{2}}\right)}{\left(a^{{\displaystyle \frac{1}{2}}}+b^{{\displaystyle \frac{1}{2}}}\right)\left(a^{\frac{1}{2}}—b^{\frac{1}{2}}\right)}=\frac{b+b^{{\displaystyle \frac{1}{2}}}{a}^{\frac{1}{2}}—b^{\frac{1}{2}}{b}^{\frac{1}{2}}}{a—b}=\frac{b+b^{{\displaystyle \frac{1}{2}}}{a}^{\frac{1}{2}}—b^{\frac{1}{2}+{\displaystyle \frac{1}{2}}}}{a—b}=\frac{b+b^{{\displaystyle \frac{1}{2}}}{a}^{\frac{1}{2}}—b}{a—b}=\frac{b^{{\displaystyle \frac{1}{2}}}{a}^{\frac{1}{2}}}{a—b}.$