<s>[INST] Evaluate $\frac{(5+6)(5^2+6^2)(5^4+6^4)\cdots(5^{1024}+6^{1024})+5^{2048}}{3^{1024}}$ Evaluate $$\frac{(5+6)(5^2+6^2)(5^4+6^4)\cdot\dots\cdot(5^{1024}+6^{1024})+5^{2048}}{3^{1024}}.$$
I can't figure out where to start. I tried using logarithms but I couldn't get a pattern going. Any advice will be helpful, thanks in advanced.
 [/INST] Hint  -  telescope:
$$
\begin{align}
&\quad \frac{\big(\color{red}{(6-5)\cdot}(6+5)\big)\;(6^2+5^2)(6^4+5^4)\cdot\dots\cdot(6^{1024}+5^{1024})+5^{2048}}{\color{red}{(6-5)\cdot}3^{1024}} \\
&= \frac{\big((6^2-5^2)(6^2+5^2)\big)\;(6^4+5^4)\cdot\dots\cdot(6^{1024}+5^{1024})+5^{2048}}{3^{1024}} \\
&= \frac{\big((6^4-5^4)(6^4+5^4)\big)\;(6^8+5^8)\cdot\dots\cdot(6^{1024}+5^{1024})+5^{2048}}{3^{1024}} \\
 &= \;\ldots
\end{align}
$$
</s>