Help needed on Special Factorizations and Clever Manipulations

1. $\color[rgb]{0.35,0.35,0.35}\frac{1}{\sqrt[3]{1}+\!\sqrt[3]{2}+sqrt[3]{4}}+\frac{1}{\sqrt[3]{4}+\!\sqrt[3]{6}+\!\sqrt[3]{9}}+\frac{1}{\sqrt[3]{9}+\!\sqrt[3]{12}+\!\sqrt[3]{16}}$

2. $\color[rgb]{.0, .184, .242}a^2+b^2=ab=4$