$\sqrt[]{\frac{8^{10}+4^{10}}{8^{4}+4^{11}}}$
= $\sqrt[]{\frac{( 2.4)^{10}+4^{10}}{( 2.4)^{4}+4^{11}}}$
= $\sqrt[]{\frac{2^{10}.4^{10}+4^{10}}{2^{4}.4^{4}+4^{11}}}$
= $\sqrt[]{\frac{2^{2^{5}}.4^{10}+4^{10}}{2^{2^{2}}.4^{4}+4^{11}}}$
= $\sqrt[]{\frac{2^{2^{5}}.4^{10}+4^{10}}{2^{2^{2}}.4^{4}+4^{11}}}$
= $\sqrt[]{\frac{4^{5}.4^{10}+4^{10}}{4^{2}.4^{4}+4^{11}}}$
= $\sqrt[]{\frac{( 4^{5}+1).4^{10}}{4^{6}+4^{11}}}$
= $\sqrt[]{\frac{( 4^{5}+1).4^{10}}{( 4^{5+1}.4^{6}}}$
= $\sqrt[]{4^{4}}= 4^{2}= 16$
