It's all about the roots

Algebra Level 4

\[\large \sqrt[3]{2-x}=1-\sqrt{x-1}\] Let \(\alpha,\beta,\gamma\) be the real roots of the equation above that satisfy the inequality \(\alpha<\beta<\gamma\). If \(\dfrac{\alpha^2+\beta^2+\gamma^2}{\gamma^2-\beta^2-\alpha^2}=\dfrac{m}{n}\), where \(m\) and \(n\) are coprime positive integers such that \(m>n\), find \[\bigg\lfloor\frac{(m.n)^{m-n}}{m+n}\bigg\rfloor.\]

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