Question

Compute
\[\huge{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^\ldots}}}}}\]

Answer 1

I'm not sure your expression makes any sense. Is the continued exponentiation finite or not?

Answer 2

It is infinite.

Answer 3

Ah, thanks for fixing it

Answer 4

Hope, you like it now. What will it be?

Answer 5

faliure

Answer 6

I'm pretty sure it's infinite.
\[\left(\left(\left(\sqrt2^\sqrt2\right)^\sqrt2\right)^\sqrt2\right)^{\ldots}\\
\left(\left(\left( \left(2^\frac{1}{2}\right) ^\sqrt2\right)^\sqrt2\right)^\sqrt2\right)^{\ldots}\\
\left(\left( \left(2^\frac{\sqrt2}{2}\right) ^\sqrt2\right)^\sqrt2\right)^{\ldots}\\
\left( \left(2^\frac{2}{2}\right)^\sqrt2\right)^{\ldots}\\
\left(2^\sqrt2\right)^{\ldots}\\
\left(\left(2^\sqrt2\right)^\sqrt2\right)^{\ldots}\\
\left(\left(\left(2^2\right)^\sqrt2\right)^\sqrt2\right)^{\ldots}\\
\left(\left(4^\sqrt2\right)^\sqrt2\right)^{\ldots}\\
\left(4^2\right)^{\ldots}\]
Continuing in this way, the base keeps getting squared.

Answer 7

The order is wrong. Compare:
\[2^{2^3}=2^8=256,\quad (2^2)^3=4^3=64.\]

Answer 8

why don't let whole of it = x ?

Answer 9

then if, you observe properly, you'll be reduced to

[ sqrt(2) ]^x = x

Answer 10

Nice, what will it be?

Answer 11

hmm, we get stuck ther! -_-

Answer 12

there*

Answer 13

2^(x/2) = x

=> 2^x = x^2

symmetrically, there is a solution , see that ?

Answer 14

integer solutions, 2 and 4...

Answer 15

which one will it be ?

Answer 16

Now tell me, what is the answer: 2 or 4?

Answer 17

i think 2

Answer 18

Why not 4?

Answer 19

because while squaring, we attached an extra root, x=4

Answer 20

Hm... \(\sqrt 2^4=4\). Why do you call it "extra root"?

Answer 21

:O .. let me think more...

Answer 22

2 also fits in though.

Answer 23

on the calculator, it approaches 2, I am not too sure how can we prove that ?

Answer 24

Let \(\epsilon > 0\)

Answer 25

A medal will be given only if you prove this.

Answer 26

-_-

Answer 27

arey to hell with the medal !

Answer 28

lol

Answer 29

LOL!!

Answer 30

Can somebody say what means "arey to hell" ?

Answer 31

ignore "arey"
I meant To Hell with the medal.

Answer 32

Ok. I found an interesting proof to this. Hope, you will find another interesting proof.

Answer 33

unable to prove, what proof you got ?

Answer 34

I mean I am unable to prove*

Answer 35

Hope, you understand.