Roots power !

By Binet's formula for Lucas numbers, $\phi^{n} + (1 - \phi)^n = L_n$ where $L_n$ is the $n\text{th}$ Lucas number. Therefore we are looking for $L_{16}.$ $L_{16} = \boxed{2207}$ If you are unfamiliar with the Lucas numbers, here is the basic definition: $L_1 = 1,\ L_2 = 3,\ L_n = L_{n-1} + L_{n-2}$ In other words, the first two numbers are $1$ and $3,$ and every number after that is the sum of the previous two numbers. They are very closely related to the much more popular Fibonacci sequence, and you can even calculate $L_n$ by adding $F_{n-1} + F_{n+1}.$

Let $a=(\frac{1+\sqrt{5}}{2}), b=(\frac{1-\sqrt{5}}{2})$ then $a+b=1$ and $ab=-1$ next $a^2+2ab+b^2=1$ therefore $a^2+b^2=-1$ doing this step many times we'll get the answer $a^{16}+b^{16}=2207$