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April 23rd, 2014, 04:07 AM   #1
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can anyone solve this??

hello...

i wanted to know if anyone can solve this :

X^X=64
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April 23rd, 2014, 04:38 AM   #2
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What is the problem you have with it? Without some idea of what you are finding difficult,, it's hard to know how to explain it.
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April 23rd, 2014, 04:42 AM   #3
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You need the Lambert W function $W$ defined as $z=W(z)e^{W(z)}$ for $z\in\mathbb C$.

Thus $\displaystyle 64\,=\,x^x$

$\displaystyle \ln64\,=\,x\ln x\,=\,(\ln x)e^{\ln x}$

So $\displaystyle W(\ln64)\,=\,\ln x$ $\displaystyle \implies$ $\displaystyle x\,=\,e^{W(\ln64)}$.
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April 23rd, 2014, 04:44 AM   #4
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There is no closed form in general.

Lambert W function - Wikipedia, the free encyclopedia

Scroll down to example 2.

I learned this from Mr. TNT (mathbalarka)

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April 23rd, 2014, 04:50 AM   #5
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anybody can find that number...it can be a "Square root" or periodic number...
the closest one is (√11.556)
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