What is the Googol root of a Googolplex?

Question

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

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

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))

Try writing it as $\sqrt[a]{b^a} = (b^a)^{1/a} = b.$ – Merkh Jun 22 '16 at 20:32 — Merkh, Jun 22 '16 at 20:32

John · Answer 1 · 2016-06-22T20:45:23.853

7

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 ...

edited Jun 22 '16 at 20:45

answered Jun 22 '16 at 20:36

John

26,319

Note that the googolth root can also be negative. – ReinstateMonica3167040 Jan 28 '20 at 00:26

score 1 · Answer 2 · answered Jun 22 '16 at 22:23

1

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

Therefore

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

answered Jun 22 '16 at 22:23

Michael Hardy

1

score 0 · Accepted Answer · answered Jun 22 '16 at 20:39

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

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

What is the Googol root of a Googolplex?

3 Answers3