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

[robo37]

... 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.

[robo37]

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).

[bo198214]

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:

\( x_L = \exp(-W_0(-\ln(y))) \)
\( x_R = \exp(-W_{-1}(-\ln(y))) \)

So when you want to get the left value - as in your case - you choose x_L and get:
\( x_L = \exp(-W_0(-\ln(3^{1/3})))\approx 2.47805268028830 \)
\( x_R \) would be simply 3 again.

[robo37]

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.

[bo198214]

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.