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

1 Answer
May 31, 2015

#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#