An algebra problem by ismet cosic

Algebra Level pending

\[\large \begin{cases} x^{z}=y^{\frac{8}{3}} \\ y^{z}=x^{\frac{2}{3}} \\ z=\sqrt[4]{x}+\sqrt[4]{y} \end{cases} \]

There are two triplets \((x,y,z)\) of solutions to this system of equations, one of which is \( (1,1,2) \). If the other set is \( (a,b,c) \), what is the value of \(\dfrac{\sqrt{a}}{b} \times c \) rounded to three decimal places?

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