Which expression is equivalent to the following, where \(a\) and \(b\) are positive? $$ a^\frac{1}{3}b^\frac{2}{3}$$

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The general form of a fractioanl exponent can be converted to an equivalent radical form:

$$ x^\frac{a}{b} = \sqrt[b]{x^a} $$

Therefore,

$$ a^\frac{1}{3}=\sqrt[3]{a^1} \text{ and } b^\frac{2}{3}=\sqrt[3]{b^2}$$

When multiplying radicals with the same radicand (number inside the radical symbol), simply multiply the terms inside.

$$ a^\frac{1}{3}b^\frac{2}{3}=\sqrt[3]{a^1}\cdot\sqrt[3]{b^2}$$ $$=\boxed{\sqrt[3]{ab^2}} $$