What is the square root of 2 to the power of 1000?

##sqrt(2)^1000 = sqrt(2)^(2xx500) = (sqrt(2)^2)^500 = 2^500##

An approximate value for this would be ##10^150## since ##2^10 = 1024 ~= 1000 = 10^3##

For a little more accuracy, use ##log_10 2 ~= 0.30103##

then ##log_10 (2^500) = 500 log_10 2 ~= 500 xx 0.30103 = 150.515##

So ##2^500 ~= 10^150.515##

Using an arbitrary precision calculator

##sqrt(2)^1000 = 2^500##

##= 327339060789614187001318969682759915221664204604306478 94832913680961337964046745548832700923259041571508866 84127560071009217256545885393053328527589376##