Oops, is it inequalities ?

Algebra Level 4

For positive real numbers \(a,b,c,d\), find the remainder when the minimum possible integer value of the following expression is divided by \(11\) \[\sqrt{(a^3+b^3+c^3+d^3)\Bigl( \dfrac{2.25}{a^3}+\dfrac{20.25}{b^3} + \dfrac{42.25}{c^3}+\dfrac{400}{d^3}\Bigr)} \]


This is a part of the set 11≡ awesome (mod remainders) .
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