mathslover 2 years ago prove that root 2 power root 2 is irrational .

1. Dido525

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2. mathslover

yep

3. Dido525

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

4. Dido525

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

5. Dido525

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

6. Dido525

7. Dido525

Assume is IS rational.

8. mathslover

ok go on

9. 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

10. Dido525

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11. ghazi

you need to use calculator

12. Dido525

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13. Dido525

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14. Dido525

This is obviously rational.

15. mathslover

^ two.. not 4

16. Dido525

Right. Still rational though.

17. ghazi

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

18. ghazi

sorry its Dido

19. mathslover

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

20. mathslover

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

21. ghazi

lol that was horrible

22. 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..

23. mathslover

^ and as well as transcendental

24. dumbcow

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

25. mathslover

yep

26. Dido525

Yeah. That's what he wanted.