A tower system

Algebra Level 4

xyxyxy..=2,yxyxyx..=32\huge x^{y^{x^{y^{x^{y^{.^{.}}}}}}} = 2 \quad,\quad y^{x^{y^{x^{y^{x^{.^{.}}}}}}} = \frac{3}{2}

Let xx and yy be real numbers such that they satisfy both the equations above. If xyxy can be expressed in the form abn\sqrt[n]{\frac{a}{b}} where gcd(a,b)=1gcd(a,b)=1 and nnis the minimum positive integer satisfying the expression, determine

LCM(a,b,n)(a+b+n) LCM(a,b,n) - (a+b+n)

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