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 (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 the indices are different, the product rule for radicals cannot be directly applied.
4. To multiply these expressions, you would need to rewrite both radicals with a common root index or convert them to exponential form using fractional 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 this does not simplify using the product rule for radicals because the bases and exponents are different.