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 states that \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\), but this rule only applies when the radicals have the same index (the same root degree).
3. Notice that the first radical is a square root (index 2) and the second is a cube root (index 3). Since their indices are different, the product rule for radicals cannot be directly applied here.
4. To multiply these expressions, you would need to rewrite them with a common root index or convert them to exponential form using rational exponents: \(\sqrt{8} = 8^{\frac{1}{2}}\) and \(\sqrt[3]{2} = 2^{\frac{1}{3}}\).
5. After rewriting, you can multiply the expressions as \$8^{\frac{1}{2}} \cdot 2^{\frac{1}{3}}$, but since the bases and exponents differ, you cannot combine them into a single radical using the product rule.