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$\sqrt[3]{\frac{8}{\sqrt{27}}}\ =\ \Bigg(\frac{8}{27^{\frac12}}\Bigg)^{\frac13}\ =\ \frac{8^{\frac13}}{27^{\frac12\cdot\frac13}}\ =\ \frac{8^{\frac13}}{27^{\frac13\cdot\frac12}}\ =\ \frac{8^{\frac13}}{(27^{\frac13})^{\frac12}}\ =\ \frac{2}{3^{\frac12}}\ =\ \frac{2}{\sqrt3}\ =\ \frac{2}{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}\ =\ \frac{2\sqrt3}{3}$

Now it is simplified in the form  $\frac{a\sqrt{b}}{c}$ .

a + b + c  =  2 + 3 + 3  =  8