Use the product rule to multiply the following.
$\sqrt{8}\cdot \sqrt[3]{2}$

• A. $\sqrt{16}$
• B. $\sqrt[3]{16}$
• C. $\sqrt[3]{10}$
• D. We cannot use the product rule.

Answer: (D)

Verified step by step guidance

1. Identify the expressions to multiply: the square root of 8, written as \(\sqrt{8}\), and the cube root of 2, written as \(\sqrt[3]{2}\).
2. Recall the product rule for radicals: \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\), which applies only when the radicals have the same index \(n\).
3. Check the indices of the radicals: the first is a square root (index 2), and the second is a cube root (index 3). Since the indices are different, the product rule cannot be directly applied.
4. To multiply these expressions, consider rewriting both radicals with a common index by expressing them as fractional exponents: \(\sqrt{8} = 8^{\frac{1}{2}}\) and \(\sqrt[3]{2} = 2^{\frac{1}{3}}\).
5. Since the bases and exponents differ, and the indices are not the same, the product rule for radicals does not apply here, so the product cannot be simplified using that rule.