Solve for

*x*, if:

3
x
x
x = 3

An algebraic solution is sought!

Substitute x^(x^(x^3)) for the three in the exponent to get:

x^(x^(x^(x^(x^(x^3))))) = 3

x^(x^(x^(x^(x^(x^(x^(x^(x^3)))))))) = 3

And repeat ad infinitum to get an infinite power tower:

x^(x^(x^(x^(x^(x^(x^(x^(x^(.........))))))))) = 3

Now the exponent of the bottom x is identical to the entire expression, so 3 can be substituted to yield x^3 = 3

The value x = cbrt(3) is the solution to all of the equations.