1. The problem asks to find an expression equivalent to $8^{\frac{1}{3}}$. 2. Recall the rule for fractional exponents: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$. 3. Here, $8^{\frac{1}{3}}$ means the cube root of 8, which can be written as $\sqrt[3]{8}$. 4. Let's analyze the options: - Option 1: $\frac{1}{\sqrt{8^3}} = \frac{1}{\sqrt{512}}$ which is not equal to $8^{\frac{1}{3}}$. - Option 2: $\sqrt{8^3} = \sqrt{512}$ which is not equal to $8^{\frac{1}{3}}$. - Option 3: $\sqrt[3]{8}$ which matches $8^{\frac{1}{3}}$. - Option 4: $\frac{1}{\sqrt[3]{8}} = 8^{-\frac{1}{3}}$ which is the reciprocal, not equal. 5. Therefore, the equivalent expression is option 3: $\sqrt[3]{8}$. Final answer: $\boxed{\sqrt[3]{8}}$