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$\text{Googol}=10^{100}$

$\text{Googolplex}=10^{\text{Googol}}=10^{{10}^{100}}$

What is $\sqrt[\text{Googol}]{\text{Googolplex}}$?

I know that's the same as $\sqrt[10^{100}]{10^{10^{100}}}$ but I still wanna know, what does this equal?

(I made this question out of idol curiosity, the only reason I did not solve this right-away is I was not thinking my own question through, after doing that I came up w/ the answer (Below))

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3 Answers

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The (positive) square root of $10^2$ is $10$.

The cube root of $10^3$ is $10$.

The googolth root of $10^{10^{100}}$ is ...

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$$\sqrt[N]x = x^{1/N}$$

Therefore

$$ \sqrt[10^{100}]{10^{(10^{100})}} = \left( 10^{\left(10^{100}\right)} \right)^{1/(10^{100})} = 10^{\Big( 10^{100} \cdot \frac 1 {10^{100}} \Big)} = 10^1 = 10. $$

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My math brain was not working to par when I posted this

Lets say you wanted to know what $\sqrt[X]{y^z}=?$

Well $\sqrt[x]{n}=n^{\frac 1x}$

Therefore $\sqrt[x]{y^z}=y^{\frac zx}$

so Replace x & z with ${10^{100}}$ & y with 10 and you get

$\sqrt[10^{100}]{10^{10^{100}}}=10^{\frac {10^{100}}{10^{100}}}=10$

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