a)
\sqrt[3]2*\sqrt[4]{2^3}*\sqrt[6]{2^4}=2^{\frac{1}{3}}*2^{\frac{3}{4}}*2^{\frac{4}{6}}=2^{\frac{4+9+8}{12}}=2^{\frac{21}{12}}=2^{\frac{7}{4}}
b)
\sqrt[4]{2\sqrt[3]{4\sqrt8}}=2^{\frac{1}{4}}*(4^{\frac{1}{3}})^{\frac{1}{4}}*((8^{\frac{1}{2}})^{\frac{1}{3}})^{\frac{1}{4}}=2^{\frac{1}{4}}*4^{\frac{1}{12}}*8^{\frac{1}{24}}=
=2^{\frac{1}{4}}*(2^2)^{\frac{1}{12}}*(2^3)^{\frac{1}{24}}=2^{\frac{1}{4}}*2^{\frac{2}{12}}*2^{\frac{3}{24}}=2^{\frac{6+4+3}{24}}=2^{\frac{13}{24}}
c)
\sqrt[5]{{2^4*\sqrt[4]2*\sqrt[6]{2^7}}}=\sqrt[5]{2^4*2^{\frac{1}{4}}*2^{\frac{7}{6}}}=\sqrt[5]{2^{4+\frac{1}{4}+\frac{7}{6}}}=
=\sqrt[5]{2^{\frac{48+3+14}{12}}}=(2^{\frac{65}{12}})^{\frac{1}{5}}=2^{\frac{65}{60}}=2^{\frac{13}{12}}
d)
\sqrt{2{\sqrt{2\sqrt{2\sqrt2}}}}=2^{\frac{1}{2}}*2^{\frac{1}{4}}*2^{\frac{1}{8}}*2^{\frac{1}{16}}=2^{\frac{8+4+2+1}{16}}=2^{\frac{15}{16}}
dodatkowe obliczenia
\sqrt{2{\sqrt{2\sqrt{2\sqrt2}}}}=2^{\frac{1}{2}}*(2^{\frac{1}{2}})^{\frac{1}{2}}*((2^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}}*(((2^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}}=2^{\frac{1}{2}}*2^{\frac{1}{4}}*2^{\frac{1}{8}}*2^{\frac{1}{16}}