mathslover
prove that root 2 power root 2 is irrational .
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Dido525
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|dw:1358647793062:dw|
mathslover
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yep
Dido525
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Well since root 2 is irrational, an irrational number raised to an irrational number should still be irrational.
Dido525
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But that's not always true. Hmm... Intresting. One moment.
Dido525
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Thing is, this is NOT an irrational number. It's rational.
Dido525
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Use proof by contradiction.
Dido525
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Assume is IS rational.
mathslover
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ok go on
ghazi
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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
Dido525
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|dw:1358648074024:dw|
ghazi
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you need to use calculator
Dido525
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|dw:1358648102171:dw|
Dido525
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|dw:1358648120754:dw|
Dido525
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This is obviously rational.
mathslover
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^ two.. not 4
Dido525
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Right. Still rational though.
ghazi
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there is a fallacy as far as i can see in your solution that you have missed square root on the exponent @djdo525
ghazi
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sorry its Dido
mathslover
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ok guys one correction, that root 2 power root 2 is rational . Sorry
But prove that root 2 power root 2 is transcendental
mathslover
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Also, PLEASE check whether it is transcendental or not , I am not sure. But I need your help.
ghazi
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lol that was horrible
mathslover
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^ and as well as transcendental
dumbcow
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@Dido525 , your example is incorrect, you showed (sqrt2)^(sqrt2)^sqrt2 is rational
but sqrt2^sqrt2 is irrational
mathslover
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yep
Dido525
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Yeah. That's what he wanted.