Simplify and rationalize the denominator: $\sqrt[3]{\frac{8}{\sqrt{27}}}.$ If the simplified expression can be expressed in the form $\frac{a\sqrt{b}}{c}$, what is $a + b + c$?
+9466 \(\sqrt[3]{\frac{8}{\sqrt{27}}}\ =\ \Bigg(\frac{8}{27^{\frac12}}\Bigg)^{\frac13}\ =\ \frac{8^{\frac13}}{27^{\frac12\cdot\frac13}}\ =\ \frac{8^{\frac13}}{27^{\frac13\cdot\frac12}}\ =\ \frac{8^{\frac13}}{(27^{\frac13})^{\frac12}}\ =\ \frac{2}{3^{\frac12}}\ =\ \frac{2}{\sqrt3}\ =\ \frac{2}{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}\ =\ \frac{2\sqrt3}{3}\)
Now it is simplified in the form \(\frac{a\sqrt{b}}{c}\) .
a + b + c = 2 + 3 + 3 = 8
+9466 \(\sqrt[3]{\frac{8}{\sqrt{27}}}\ =\ \Bigg(\frac{8}{27^{\frac12}}\Bigg)^{\frac13}\ =\ \frac{8^{\frac13}}{27^{\frac12\cdot\frac13}}\ =\ \frac{8^{\frac13}}{27^{\frac13\cdot\frac12}}\ =\ \frac{8^{\frac13}}{(27^{\frac13})^{\frac12}}\ =\ \frac{2}{3^{\frac12}}\ =\ \frac{2}{\sqrt3}\ =\ \frac{2}{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}\ =\ \frac{2\sqrt3}{3}\)
Now it is simplified in the form \(\frac{a\sqrt{b}}{c}\) .
a + b + c = 2 + 3 + 3 = 8
