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If I am given something like $\sqrt[n]{100} = 10$ then it is obvious that $n=2$.

But say I get something like $\sqrt[n]{2} = 36$; how do I find what $n$ is then?

Ottavio
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user577111
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1 Answers1

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Fleshing out Jet Chung's comment, writing $\sqrt[n]2$ as $2^{1/n}$, we have

$$2^{1/n}=36\iff{1\over n}\log2=\log36\iff n={\log2\over\log36}$$

Note, it doesn't matter what base you use for the logarithm, the quotient is the same. Note also, the $n$th root notation $\sqrt[n]{x}$ is usually reserved for positive integer values of $n$, which is not the case here. (Since $36\gt32=2^5$, it's easy to see that $n\lt1/5$.)

Barry Cipra
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