Exponent Rules Question 3

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

\sqrt[3]{ab^2}

\sqrt{ab^2}

\dfrac{1}{\sqrt[3]{ab^2}}

\dfrac{1}{\sqrt{ab^2}}

Approach

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}}