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Numbers are often used to define more complicated numbers. For example, twointegers are used to define a rational number and two reals are used to definea complex number. It might be expected that an irrational power of anirrational number would be an irrational number. Despite that, it is actuallypossible for an irrational power of an irrational number to be a rationalnumber. For many generations, this circulated as a non-constructive proof bycontradiction in logic for discrete mathematics textbooks and college courses.Since the 80s, a constructive proof circulated orally, such as$\sqrt{2}^{log_\sqrt{2} 3}$ equals to 3. A written proof was published in 2008by Lord. The first contribution of this paper is to prove that there is anuncountable number of such pairs of irrational numbers such that the power ofone to the other is a rational number. Marshall and Tan answered the questionof whether there is a single irrational number $a$ such that $a^a$ is rational.They proved that given $I$ = $(\frac{1}{e})^{(\frac{1}{e},\infty)}$, then everyrational number in $I$ is either of the form $a^a$ for an irrational $a$ or isin the very thin set $\{1,4,27,256,...,n^n,...\}$. It seems a challenging taskto analytically solve the equation $x^x=y$ for any real $y$. To the best of ourknowledge, there is no work on finding $x$ from a given $y$. We proved that$ln((ln(y))<x$ for $y>e$ and $x<ln(y)$, for $y> e^e$. Hence, the secondcontribution of this paper is to estimate the real $x$ such that either one ofthe equations $x^x=y$ or $x^{x^x}=y$ holds, for a given $y$.

Proving the uncountability of the number of irrational powers of  irrational numbers evaluated as rationals and solutions approximation for  x^x=y and x^{x^x}=y