$\sqrt[5]{8\sqrt{2\sqrt[3]{2}}}$

$=\sqrt[5]{8\sqrt{2.2^\dfrac{1}{3}}}$

$=\sqrt[5]{8\sqrt{2^\dfrac{4}{3}}}$

$=\sqrt[5]{2^3.2^\dfrac{2}{3}}$

$=\sqrt[5]{2^\dfrac{11}{3}}$

$=2^\dfrac{11}{15}$

`=>m=11;n=15`

`=>P=m^2+n^2=346`

`=>D`