It's really a matter of simplification

Algebra Level 3

\[ \large \frac{\left(x+\dfrac{1}{x}\right)^6 - \left(x^6 + \dfrac{1}{x^6}\right) - 2}{\left(x+\dfrac{1}{x}\right)^3 + \left(x^3+\dfrac{1}{x^3}\right)} \]

Find the minimum value of the expression above for all \( x>0\).


Note: This particular problem appeared in the 1998 Putnam Math Competition.
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