Calculator-killer-2

On the left hand side of the equation, we rewrite the denominator as a power of 2. The denominator is $\displaystyle 16^{2} = (2^{4})^{2} = 2^{8}$ , so the fraction $\frac{2^{16}}{(16)^{2}}$ cancels out to $2^{8}$ .

We take the logarithm with base 2 on both sides of the equation $\displaystyle 2^{8} = 2^{2^{x}}$ , giving $2^{x} = 8$ .

Taking the logarithm with base 2 again, we end up with $\boxed {x=3}$ .

Note: exponent towers are evaluated from the top, down. Please see this page: https://brilliant.org/wiki/what-is-a-to-the-b-to-the-c/

$\frac{2^{16}}{16^2}=\frac{2^16}{2^8}=2^8 \\ 2^8=2^{2^x}\Rightarrow x=3$

As easy as calculating powers of 2. $2^{16}=65536=10000_{16}$ and $16^2=100_{16}$ so the quotient becomes $100_{16}=2^8$ so $x=3$