$1) \frac{y—16y^{{\displaystyle \frac{1}{2}}}}{5y^{{\displaystyle \frac{1}{4}}}+20}.$

Воспользуемся свойствами степеней

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

и формулой сокращенного умножения

$a^2—b^2=\left(a—b\right)\left(a+b\right).$ $\frac{y—16y^{{\displaystyle \frac{1}{2}}}}{5y^{{\displaystyle \frac{1}{4}}}+20}=\frac{y^{{\displaystyle \frac{1}{2}+\frac{1}{2}}}—16y^{{\displaystyle \frac{1}{2}}}}{5y^{{\displaystyle \frac{1}{4}}}+20}=\frac{y^{{\displaystyle \frac{1}{2}}}·y^{\frac{1}{2}}—16y^{{\displaystyle \frac{1}{2}}}}{5y^{{\displaystyle \frac{1}{4}}}+20}=\frac{y^{{\displaystyle \frac{1}{2}}}\left(y^{\frac{1}{2}}—16\right)}{5\left(y^{{\displaystyle \frac{1}{4}}}+4\right)}=\frac{y^{{\displaystyle \frac{1}{2}}}\left(y^{2·\frac{1}{4}}—4^2\right)}{5\left(y^{{\displaystyle \frac{1}{4}}}+4\right)}=\frac{y^{{\displaystyle \frac{1}{2}}}\left({\left(y^{\frac{1}{4}}\right)}^2—4^2\right)}{5\left(y^{{\displaystyle \frac{1}{4}}}+4\right)}=\frac{y^{{\displaystyle \frac{1}{2}}}\left(y^{\frac{1}{4}}+4\right)\left(y^{\frac{1}{4}}—4\right)}{5\left(y^{{\displaystyle \frac{1}{4}}}+4\right)}=\frac{y^{{\displaystyle \frac{1}{2}}}\left(y^{\frac{1}{4}}—4\right)}{5}.$ $2) \frac{a^{{\displaystyle \frac{4}{5}}}—b^{{\displaystyle \frac{4}{5}}}}{a^{{\displaystyle \frac{2}{5}}}—b^{{\displaystyle \frac{2}{5}}}}.$

Воспользуемся свойством степеней

${\left(a^n\right)}^m=a^{n·m}$

и формулой сокращенного умножения

$a^2—b^2=\left(a—b\right)\left(a+b\right).$ $\frac{a^{{\displaystyle \frac{4}{5}}}—b^{{\displaystyle \frac{4}{5}}}}{a^{{\displaystyle \frac{2}{5}}}—b^{{\displaystyle \frac{2}{5}}}}=\frac{a^{{\displaystyle 2·\frac{2}{5}}}—b^{{\displaystyle 2·\frac{2}{5}}}}{a^{{\displaystyle \frac{2}{5}}}—b^{{\displaystyle \frac{2}{5}}}}=\frac{{\left(a^{{\displaystyle \frac{2}{5}}}\right)}^2—{\left(b^{{\displaystyle \frac{2}{5}}}\right)}^2}{a^{{\displaystyle \frac{2}{5}}}—b^{{\displaystyle \frac{2}{5}}}}=\frac{\left(a^{\frac{2}{5}}—b^{\frac{2}{5}}\right)\left(a^{\frac{2}{5}}+b^{\frac{2}{5}}\right)}{a^{{\displaystyle \frac{2}{5}}}—b^{{\displaystyle \frac{2}{5}}}}=a^{\frac{2}{5}}+b^{\frac{2}{5}}.$