Simplifying Large Powers

$\large\sqrt{\dfrac{8^{10}+4^{10}}{8^4+4^{11}}}$

= $\large\sqrt{\dfrac{2^{30}+2^{20}}{2^{12}+2^{22}}}$

= $\large\sqrt{\dfrac{2^{12}(2^{18}+2^8)}{2^{12}(1+2^{10})}}$

= $\large\sqrt{\dfrac{2^8(2^{10}+1)}{2^{10}+1}}$

= $\large\sqrt{2^8}$

= $2^4$

= $\boxed{16}$

$\sqrt{\frac {8^{10}+4^{10}}{8^4+4^{11}}} = \sqrt{\frac {2^{30}+2^{20}}{2^{12}+2^{22}}} =\sqrt{\frac {2^{20}\left(2^{10}+1\right)}{2^{12}\left( 1 +2^{10}\right)}} = \sqrt{\frac {2^{20}}{2^{12}}} = \sqrt{2^8} = 2^4 = \boxed{16}$

An alternative to breaking down all terms to powers of 2 is to rewrite the powers of 8 as

$8^{10} = 2^{10} 4^{10} = 4^5 4^{10} = 4^{15}$

$8^{4} = 2^{4} 4^{4} = 4^{2}4^{4} = 4^{6}$

Then

$\sqrt{ \frac{8^{10} + 4^{10}} {8^{4} + 4^{11}} } =\sqrt{ \frac{4^{15} + 4^{10}} {4^{6} + 4^{11}} } =\sqrt{ \frac{4^{10}(4^{5}+1)} {4^{6} (1+ 4^{5})} } =\sqrt{ \frac{4^{10}} {4^{6}} } =\sqrt{ 4^{4} } =4^{2} = 16$