... 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.
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e is the global maximum of x root x, 2 root 2 = 4 root 4, so...

02/13/2011, 12:30 AM
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).
02/13/2011, 10:27 AM
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.
Thank you very much for the reply.
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.
02/15/2011, 05:08 PM
(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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