Let $t$ be a positive real number such that $t^4 + t^{-4} = 2023$. Determine the value of $t^3 + t^{-3}$ in the form of $a\sqrt b$, where $a$ and $b$ are positive integers.
Amir Hossein wrote:
Let $t$ be a positive real number such that $t^4 + t^{-4} = 2023$. Determine the value of $t^3 + t^{-3}$ in the form of $a\sqrt b$, where $a$ and $b$ are positive integers.
Let $s_k=t^k+t^{-k}$
We have $s_4=2023$
So $s_2^2=s_4+2=2025$ and $s_2=45$
So $s_1^2=s_2+2=47$ and $s_1=\sqrt{47}$
And $s_1s_2=s_3+s_1$ gives $\boxed{s_3=44\sqrt{47}}$