Sixteens!

Rewriting this equation:

$\sqrt { { 16 }^{ 2 }+{ 16 }^{ 2 }+{ 16 }^{ 2 }+...+{ 16 }^{ 2 }+{ 16 }^{ 2 }+{ 16 }^{ 2 } }={16}^{16}$ becomes $\sqrt {n\cdot{16}^{2}}={16}^{16}$ .

Solving:

$\sqrt {n\cdot{16}^{2}}={16}^{16}$

$16\sqrt { n } ={ 16 }^{ 16 }$

$\sqrt { n } ={ 16 }^{ 15 }$

$n={ 16 }^{30 }$

Let the number of times $16^{2}$ comes under the square root be $x$ . Then, $\sqrt {x \times 16^{2}} = 16^{16} \Rightarrow x \times 16^{2} =(16^{16})^{2}=16^{32} \Rightarrow x = \dfrac{16^{32}}{16^{2}} = \boxed{16^{30}}$ .

Let x be the number of times 16^2 should appear. We find x to be equal to 16^15. Squaring it we get 16^30