<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] First we simplify the numerator of $$ \frac{(5+6)(5^2+6^2)(5^4+6^4)\cdot\dots\cdot(5^{1024}+6^{1024})+5^{2048}}{3^{1024}}.$$
$$(5+6)(5^2+6^2)=(6-5)(5+6)(5^2+6^2)=(6^4-5^4)$$
$$(5+6)(5^2+6^2)(5^4+6^4)=(6^4-5^4)((5^4+6^4)=(6^8-5^8)
\\.\\.\\ (5+6)(5^2+6^2)(5^4+6^4)\cdot\dots\cdot(5^{1024}+6^{1024})=(6^{2048}-5^{2048})$$
$$ \frac{(5+6)(5^2+6^2)(5^4+6^4)\cdot\dots\cdot(5^{1024}+6^{1024})+5^{2048}}{3^{1024}}=\frac {  6^{2048} }{ 3^{1024}} = 12^{1024} $$
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