prove that root 2 power root 2 is irrational .

- mathslover

prove that root 2 power root 2 is irrational .

- schrodinger

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- anonymous

|dw:1358647793062:dw|

- mathslover

yep

- anonymous

Well since root 2 is irrational, an irrational number raised to an irrational number should still be irrational.

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- anonymous

But that's not always true. Hmm... Intresting. One moment.

- anonymous

Thing is, this is NOT an irrational number. It's rational.

- anonymous

Use proof by contradiction.

- anonymous

Assume is IS rational.

- mathslover

ok go on

- ghazi

i dont have a very nice method but i guess this will help you let's say \[x= 2 ^{\sqrt2}\] take log on both the sides \[\log x = \sqrt {2}\log2\] find the value of right side and finally \[X= e ^{\sqrt2 \log 2}\] and i am sure that would be irrational

- anonymous

|dw:1358648074024:dw|

- ghazi

you need to use calculator

- anonymous

|dw:1358648102171:dw|

- anonymous

|dw:1358648120754:dw|

- anonymous

This is obviously rational.

- mathslover

^ two.. not 4

- anonymous

Right. Still rational though.

- ghazi

there is a fallacy as far as i can see in your solution that you have missed square root on the exponent @djdo525

- ghazi

sorry its Dido

- mathslover

ok guys one correction, that root 2 power root 2 is rational . Sorry
But prove that root 2 power root 2 is transcendental

- mathslover

Also, PLEASE check whether it is transcendental or not , I am not sure. But I need your help.

- ghazi

lol that was horrible

- mathslover

OK friends I am totally confused. http://www.math.hmc.edu/funfacts/ffiles/30002.3-5.shtml
Here I got that root 2 power root 2 is irrational..

- mathslover

^ and as well as transcendental

- dumbcow

@Dido525 , your example is incorrect, you showed (sqrt2)^(sqrt2)^sqrt2 is rational
but sqrt2^sqrt2 is irrational

- mathslover

yep

- anonymous

Yeah. That's what he wanted.

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