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e is the global maximum of x root x, 2 root 2 = 4 root 4, so...
#1
... what when put to the root of itself is equal to the cube root of 3? Can it be expressed in terms of e? I know this sounds like a bit of a random question but it's something I've always been curious in.

[Image: 512pxxthrootofxsvg.png]
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#2
I've calculated that the number is roughly 2.47805268, but I don't think I can calculate a way to express it without learning advanced mathematics (as you can probably tell I'm no mathematician).
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#3
Oh, now with the picture I see what you mean.

To get a formula we can use the Lambert W function.
The Lambert W function is the inverse of the function M(x)=x*e^x
and we can express the self root x^(1/x) with help of M:

y = x^(1/x) = exp(-M(-ln(x)))

You can verify this with a bit of calculation.
Then we can obtain the inverse:

exp(-W(-ln(y))) = x

More exactly x can be two values, left and right from e, which correspond to the two branches of W:




So when you want to get the left value - as in your case - you choose x_L and get:

would be simply 3 again.
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#4
Thank you very much for the reply. Smile

So, if you take the Lambert W function out of the equation, what are you left with? As I said I'm no mathematician I'm afraid so I don't fully understand the principle.
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#5
(02/15/2011, 03:20 PM)robo37 Wrote: So, if you take the Lambert W function out of the equation, what are you left with? As I said I'm no mathematician I'm afraid so I don't fully understand the principle.

I dont think there might be a closed form solution without the Lambert W function. However if you are after calculating the number, that function really helps as it is available in most computer algebra packages.

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