Radical Dude!

Algebra Level 5

527152715278=a+bc,\large \sqrt[8]{527 - \dfrac{1}{527 - \frac{1}{527 - \cdots}}} = \dfrac{\sqrt{a} + \sqrt{b}}{c} ,

where a,b,ca,b,c are positive, square-free integers. Find the product abcabc.

Details and Assumptions:

  • To clarify, it is the 8th8^\text{th} root that is being taken.
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