1. Simplify $\sqrt[2003]{2 \sqrt{11}-3 \sqrt{5}} \cdot \sqrt[4006]{89+12 \sqrt{55}}$
Solution: $\quad-1$
Note that $(2 \sqrt{11}+3 \sqrt{5})^{2}=89+12 \sqrt{55}$. So, we have
$$
\begin{aligned}
\sqrt[2003]{2 \sqrt{11}-3 \sqrt{5}} \cdot \sqrt[4006]{89+12 \sqrt{55}} & =\sqrt[2003]{2 \sqrt{11}-3 \sqrt{5}} \cdot \sqrt[2003]{2 \sqrt{11}+3 \sqrt{5}} \\
& =\sqrt[2003]{(2 \sqrt{11})^{2}-(3 \sqrt{5})^{2}}=\sqrt[2003]{-1}=-1
\end{aligned}
$$