<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:
$$\frac{(5+6)+5^2}{3^1}=12$$
$$\frac{(5+6)(5^2+6^2)+5^4}{3^2}=144=12^2$$
$$\frac{(5+6)(5^2+6^2)(5^4+6^4)+5^8}{3^4}=20736=12^4$$
There is a pattern there. See if you can prove that the pattern continues.

One way is to generalize how something like $(5+6)(5^2+6^2)+5^4$ squares to become equal to $(5+6)(5^2+6^2)(5^4+6^4)+5^8$. [Because then just square both sides of one of the above equations to get to the next equation.]
As in:
$$
\begin{align}
\left((5+6)(5^2+6^2)+5^4\right)^2&=(5+6)^2(5^2+6^2)^2+2(5+6)(5^2+6^2)5^4+5^8\\
&=(5+6)(5^2+6^2)\left[(5+6)(5^2+6^2)+2(5^4)\right]+5^8
\end{align}
$$
So now it is about generalizing that something like $(5+6)(5^2+6^2)+2(5^4)$ is the same as $(5^4+6^4)$. I think writing all the $6$'s as $5+1$ and using the binomial theorem might get you there.
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