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

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

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Answer 1 · 2015-05-31T08:19:09

${\sqrt{2}}^{1000} = {\sqrt{2}}^{2 \times 500} = {({\sqrt{2}}^{2})}^{500} = 2^{500}$ $sqrt(2)^1000 = sqrt(2)^(2xx500) = (sqrt(2)^2)^500 = 2^500$

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

For a little more accuracy, use ${log}_{10} 2 ≅ 0.30103$ $log_10 2 ~= 0.30103$

then ${log}_{10} (2^{500}) = 500 {log}_{10} 2 ≅ 500 \times 0.30103 = 150.515$ $log_10 (2^500) = 500 log_10 2 ~= 500 xx 0.30103 = 150.515$

So $2^{500} ≅ 10^{150.515}$ $2^500 ~= 10^150.515$

Using an arbitrary precision calculator

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

$= 327339060789614187001318969682759915221664204604306478 94832913680961337964046745548832700923259041571508866 84127560071009217256545885393053328527589376$

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

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