\[\sqrt{x^{x^{x...}}\sqrt{x^{x^{x...}}\sqrt{x^{x^{x...}}...}}}^{\sqrt{x^{x^{x...}}\sqrt{x^{x^{x...}}\sqrt{x^{x^{x...}}...}}}^{\sqrt{x^{x^{x...}}\sqrt{x^{x^{x...}}\sqrt{x^{x^{x...}}...}}}^{.^{.^{.}}}}}=16\]

\[ x^{16 {\sqrt[16]{16}}} =?\]

DETAILS AND ASSUMPTIONS:

x is a positive real number.

Look only for positive, real solutions.

Acknowledgement:

The original problem was incorrectly interpreted by the solvers; the expression is exponentiated into itself infinitely many times, not just three (note the dot-dot-dot on top). Wrong answers were entered, but were accepted by the system as correct (because I originally solved my own problem incorrectly).

I apologize for the mishap.

(the exponent reads "16 times the 16th root of 16")

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