How would one simplify $5^{\sqrt5}$? A constant raised to a radical. Please show steps. On a calculator I get $36.554$.. Thanks you. Trying to see how it would be done on paper not just a calculator.
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What do you mean "solve"? Find the value? Simplify? – internet_user Feb 05 '18 at 02:00
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Yes my bad find the value – user522534 Feb 05 '18 at 02:01
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It doesn't get "simpler" than that (but you can simplify $\sqrt 5^5$) – Akababa Feb 05 '18 at 02:11
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Are you looking for how to simpliy it or how to get 36.544... by hand? – Ovi Feb 05 '18 at 02:12
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@Ovi looking for both – user522534 Feb 05 '18 at 02:12
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Compute or use tables to find the logarithm $$x = \log_{10}5^{\sqrt{5}}$$ $$x = \sqrt{5} \log_{10}5$$ Then compute or use tables to take the inverse logarithm $$5^{\sqrt{5}} =10 ^x$$
Andy Walls
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"Compute" and he just has paper and pencil. How could he ever find $x$ in the last step without the use of a calculator, please explain that. – Yash Jain Feb 05 '18 at 02:18
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@YashJain See this answer https://math.stackexchange.com/questions/61279/calculate-logarithms-by-hand – Andy Walls Feb 05 '18 at 02:33
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@user522534 Computing the inverse log (base 10) of $\sqrt{5}\log_{10}5$ works on my calculator. – Andy Walls Feb 05 '18 at 02:37
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(+1) I think this is the correct way. Besides, as for this particular case, $\log 5$ is a decent memorable value (along with $\log 2$, $\log 3$ and $\log 7$) – Gaurang Tandon Feb 05 '18 at 02:39
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@AndyWalls so if i take sqrt(5)Log(5) I get 1.562944.. so now ur saying take the inverse? Im confused – user522534 Feb 05 '18 at 02:59
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@user522534 yes. The inverse log (base 10) of 1.562944 is $10^{1.562944} = 36.544..$ – Andy Walls Feb 05 '18 at 03:05