# Positive Rational Number as Power of Number with Power of Itself

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## Theorem

Every positive rational number can be written either as:

- $a^{a^a}$ for some irrational number $a$

or as:

- $n^{n^n}$ for some natural number $n$.

## Proof

## Sources

- March 2012: J. Marshall Ash and Yiren Tan:
*A rational number of the form $a^a$ with $a$ irrational*(*The Mathematical Gazette***Vol. 96**: 106 – 109)