Cube Roots In The Denominators Of Fractions

How can I find the sum of $\frac{1}{\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}}+\frac{1}{\sqrt[3]{4}+\sqrt[3]{6}+\sqrt[3]{9}}+\frac{1}{\sqrt[3]{9}+\sqrt[3]{12}+\sqrt[3]{16}}$
$(\mathrm{M}\mathrm{A}\Theta\:1992)$