And then we say that it's impossible!

Calculus Level 3

$x$ is a non-negative real number which satisfies the following equation.

$\huge x^{x^{x^{x^{.^{.^{x^{2}}}}}}}=2$

Then find the value of $x$ . Correct your answers to three decimal places.

Enter 666 if you come to the conclusion that no such $x$ exists.

1 solution

Abhay Tiwari
May 6, 2016

Since we are not sure of the convergence of the power tower so to make it easy we should not touch the L.H.S. and let's take the R.H.S.

$\large 2={\sqrt[2]{2}}^{2}$

$\large 2={\sqrt[2]{2}}^{{\sqrt[2]{2}}^{2}}$

$\huge 2={\sqrt[2]{2}}^{{\sqrt[2]{2}}^{{\sqrt[2]{2}}^{2}}}$

$\huge 2={\sqrt[2]{2}}^{{\sqrt[2]{2}}^{{\sqrt[2]{2}}^{.^{.^{.^{2}}}}}}$

$x=\sqrt[2]{2}=\boxed{1.414}$

I think that writing $\huge x^{x^{.^{.^{.^{x^2}}}}}$ is more clear.

展豪張 - 5 years, 1 month ago

Yes, I guess it looks good for now :)

Abhay Tiwari - 5 years, 1 month ago

Haha that's great! :)

展豪張 - 5 years, 1 month ago