Yet another question 5

Algebra Level 5

Let a quadratic equation be represented as

A(32)x2+Bx(3+2)+C=0,A(\sqrt{3} - \sqrt{2})x^2 + \dfrac {Bx}{(\sqrt{3} + \sqrt{2})} + C=0,

where A=(49+206)1/4A = (49 + 20\sqrt 6)^{1/4} and B=83+863+163+....B= 8\sqrt 3 + \dfrac{8\sqrt6}{\sqrt 3} + \dfrac{16}{\sqrt3} + .....

Let α\alpha and β\beta be the roots of the above equation which are related to the constraint (αβ=(66)k,( \displaystyle | \alpha - \beta | = (6\sqrt6)^k,

where k=log6102log65+log6log618+log672\displaystyle k = \log_{6}10 - 2\log_6 \sqrt5 + \log_6\sqrt{\log_{6} 18 + \log_{6} 72}.

Find the value of CC.

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