On a recent Giant Bombcast, someone wrote in and asked an absurd question (as is usual for this podcast). In short, the question was:
Given a 1080p TV, how long would it take to view every possible image?
(The guys on the podcast eventually concluded that it would take longer than the lifespan of the universe. On the next episode, someone else wrote in and said that it would take "the lifetime of the universe infinity times over". However, this is not the focus of the question.)
This question effectively boils down to:
$$ n = (256^3) ^ {1920 \times 1080} $$
Or
$$ n = 16777216^{2073600} $$
This is obviously an incomprehensibly large number. I suspect that it has billions of digits and such. However, is there any way to figure out any properties of it without having to actually calculate it? Like, is it possible to figure out what it would be in scientific notation? Or something like that?
This 14,981,180-digit number is $2^{49766400}$ which is $1.50041692264871365956211935\ldots\times10^{14981179}$ in scientific notation. Its final digits are ...26416858621186596199148240803900704079262350188482842853376.
I'll try to put this in human terms, but it's not going to make it any easier. This is roughly the number of ways to choose a team of four million people from all of the earth.