A tower system

Since $x^{y^{x^\ldots}} = 2$ and $y^{x^{y^\ldots}}=\dfrac{3}{2}$ , we have

$x^{\frac{3}{2}} = 2$

$y^{2} = \frac{3}{2}$

by substitution. Simplifying and converting, we have

$x = 2^{\frac{2}{3}} = 2^{\frac{4}{6}} = \sqrt[6]{16}$

$y=\left (\dfrac{3}{2} \right )^{\frac{1}{2}} = \left (\dfrac{3}{2} \right )^{\frac{3}{6}} = \sqrt[6]{\dfrac{27}{8}}$

Therefore,

$xy = \sqrt[6]{16} \times \sqrt[6]{\dfrac{27}{8}} = \sqrt[6]{54}$

Hence, $a=54, b=1, c=6$ .

$LCM(54, 1, 6) = 54$

$54-(54+6+1) = -7$

Therefore, the answer is $-7$ .

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The reason I converted the fractions in the exponents into like fractions is so that it would be easy to multiply out the expression in the end.

$x^\frac{3}{2}=2$ and $y^2=\frac{3}{2}$ upon simplifying $x×y$ we will have $\sqrt[6]{54}=xy$

Now $a=54,b=1,c=6$ $L.C.M (54,1,6)=54$

Now simply calculate $54-(54+6+1)$ which is $\boxed{-7}$

Same solution lang kami ni Kesa