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if a and b are irrational, does it follow that a^b is irrational?
Consider: 1) sqrt2^sqrt2
2) (sqrt2^sqrt2)^sqrt2
5 Answers
- HikerLv 41 decade agoFavorite Answer
Given a and b are irrational, a^b could be rational. The interesting thing about your second expression to consider is, given (sqrt 2 ^ sqrt 2) ^ sqrt 2,
we know it is equal to (sqrt 2) ^2 by multiplying exponents. This gives us a value of 2.
What we still don't know in the example is whether or not (squrt 2 ^ sqrt 2) is rational or irrational.
- BenLv 61 decade ago
Hiker, you've just about got it. We know that expression (2) is rational, right? We don't really know if sqrt(2)^sqrt(2) is rational, as you mentioned. If it's irrational, then we have (2) is irrational^irrational=rational, which contradicts the hypothesis. But if it's rational, then expression (1) is a contradiction. So the answer to the original question would be no.
EDIT: steiner1745 answers the question of whether (1) is rational...nicely done. I remembered that it was indeed irrational, but had never seen a reason behind it. For the purposes of showing that a^b need not be irrational even when a and b are irrational though, we don't need to know this bit.
Oh, and steiner1745, when I was first presented with this proposition a few months ago, e^(i*pi) was my first response too. My instructor, a graph theorist, didn't like my use of complex numbers :)
- steiner1745Lv 71 decade ago
Consider
e^(iÏ) = -1.
e and iÏ are certainly irrational, so the answer to
your question is no.
BTW. A poster asked whether â2^â2 is rational.
This number is not only irrational, it is transcendental.
This follows from the theorem of Gelfond-Schneider,
which states that a^b is transcendental if
1). a is algebraic, a .ne. 0 or 1
and
2. b is algebraic and irrational.
Since â2 is a root of x²-2 = 0 and â2 is irrational,
the result follows.
- jared_e42Lv 51 decade ago
No. The first example proves that 2#s that are irrational can be multiplied to be a rational #. Therefore, it does not logically follow.
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- Helen BLv 51 decade ago
sqrt(2)^sqrt(2) = 1.6325....
(sqrt(2)^sqrt(2))^sqrt(2) = 2 - that's what my calculator says, I don't know how to prove it.