Sum This!

Algebra Level 2

113+23+43+143+63+93+193+123+163 \Large{\frac1{\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}} + \frac1{\sqrt[3]{4}+\sqrt[3]{6}+\sqrt[3]{9}} + \frac1{\sqrt[3]{9}+\sqrt[3]{12}+\sqrt[3]{16}}}

If the expression above can be simplified to the form of a3+b{\sqrt[3] a + b} for integers aa and bb, find the value of a+ba+b.

Source: MAΘMA\Theta 1992.
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