Prove This If You Can

If x is a positive rational number, show that ,

${ x }^{ x }$

is irrational unless x is an integer.

#NumberTheory #Exponents #IrrationalNumbers

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Let x = a/b, and x^x = c/d, where a, b, c, d, are integers, such that a & b and c & d are co-primes. Then we have after some re-arranging:

(a^a) (d^a) = (b^b) (c^b)

Since both a & b and c & d are co-primes, a^a = c^b and d^a = b^b, neither which is possible because a & b are co-primes, unless b = 1 and d = 1.

Michael Mendrin - 7 years, 3 months ago

Ow, nice! The only thing that I think your proof lack is LaTeX. Anyway, thank you.

Damiann Mangan - 7 years, 3 months ago

Yeah, I should learn how to us LaTeX, you're right about that.

Michael Mendrin - 7 years, 3 months ago

@Michael Mendrin – hey michael you can use $Daum equation editor$

Rishabh Jain - 7 years, 1 month ago

@Rishabh Jain – wen I used this Daum eqn editor , I wasn't able to copy the equation and paste it in the Field.. It just copies the code only !! Any solution ??

Sandeep KV - 6 years, 10 months ago

It's right to assume that a,b,c,d are relative prime?

Paul Ryan Longhas - 6 years, 8 months ago

Here you go @Michael Mendrin:

Let $x=\frac{a}{b}$ and $x^x=\frac{c}{d}$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a$ and $b$ are coprime as well as $c$ and $d$ . After some rearranging, we attain

$(a^a)(d^a)=(b^b)(c^b)$

Since both $a$ and $b$ are coprime, $a^a=c^b$ , and $d^a=b^b$ . Neither of these statements are possible, because $a$ and $b$ are coprime, unless the case where $b=1$ and $d=1$ .

Finn Hulse - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$