How I can calculate this number

Question

How I can calculate exactly this number:

$$ \frac{(\sqrt{3}+2)^{2^{333111366}}+1}{(2^{333111367}-1)(\sqrt{3}+2)^{2^{333111365}}} $$

Several softwares do not support this number.

Answer 1

$$\frac{(2+\sqrt3)^{2^{333111366}}+1}{(2^{333111367}-1)(2+\sqrt{3})^{2^{333111365}}}$$If this is Your Number, The Wolfram Alpha says It's

$$10^{10^{10^{8.001199221375904}}}$$

and it also says that It's not a positive integer

Answer 2

The number is so large it cannot be reasonably represented. I ignored the $+1, -1$ and took the log in Alpha. The result is a little over $10^{10^8}$, so in decimal would have more than that many digits. Note that $3+\sqrt 2 \gt 2$ and dividing the two terms involving it leaves (ignoring the $+1$ in the numerator) $3^{2^{333111365}}$ which completely swamps the puny power of $2$ in the denominator, which only has about $10^8$ digits