Evaluate. No Need for Calculators

To solve $\sqrt[2018]{3\sqrt{5} - 2\sqrt{11}} \times \sqrt[4036]{89+12\sqrt{55}}$

Note that $(3\sqrt{5}+2\sqrt{11})^2=89+12\sqrt{55}$

then $\sqrt[2018]{3\sqrt{5} - 2\sqrt{11}} \times \sqrt[4036]{89+12\sqrt{55}} = \sqrt[2018]{3\sqrt{5} - 2\sqrt{11}} \times \sqrt[4036]{(3\sqrt{5}+2\sqrt{11})^2}$

$\sqrt[2018]{(3\sqrt{5} - 2\sqrt{11})} \times \sqrt[2018]{(3\sqrt{5}+2\sqrt{11})}= \sqrt[2018]{(3^2\times 5) - (2^2\times 11)} = \sqrt[2018]{45-44} = \boxed{1}$

$\begin{aligned} \sqrt[2018]{3\sqrt 5-2\sqrt{11}} \times \sqrt[4036]{89+12\sqrt{55}} & = \sqrt[4036]{\left(3\sqrt 5-2\sqrt{11}\right)^2} \times \sqrt[4036]{89+12\sqrt{55}} \\ & = \sqrt[4036]{\left(45-12\sqrt{55} + 44\right) \left(89+12\sqrt{55}\right)} \\ & = \sqrt[4036]{\left(89-12\sqrt{55}\right) \left(89+12\sqrt{55}\right)} \\ & = \sqrt[4036]{89^2-12^2\times 55} \\ & = \sqrt[4036]{7921-7920} \\ & = \boxed{1} \end{aligned}$