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

May 31, 2015

${\sqrt{2}}^{1000} = {\sqrt{2}}^{2 \times 500} = {\left({\sqrt{2}}^{2}\right)}^{500} = {2}^{500}$

An approximate value for this would be ${10}^{150}$ since ${2}^{10} = 1024 \cong 1000 = {10}^{3}$

For a little more accuracy, use ${\log}_{10} 2 \cong 0.30103$

then ${\log}_{10} \left({2}^{500}\right) = 500 {\log}_{10} 2 \cong 500 \times 0.30103 = 150.515$

So ${2}^{500} \cong {10}^{150.515}$

Using an arbitrary precision calculator

${\sqrt{2}}^{1000} = {2}^{500}$

= 327339060789614187001318969682759915221664204604306478 94832913680961337964046745548832700923259041571508866 84127560071009217256545885393053328527589376