Use the product rule to multiply the following.
$\sqrt[4]{7m^2}\cdot \sqrt[4]{2n}$

• A. $\sqrt[4]{7m^2+2n}$
• B. $\sqrt[4]{14m^{2}n}$
• C. $\left(\sqrt[4]{14m^2}\right)n$
• D. $\sqrt[4]{7m^2}+\sqrt[4]{2n}$

Answer: (B)

Verified step by step guidance

1. Recognize that the problem involves multiplying two fourth roots: \(\sqrt[4]{7m^2}\) and \(\sqrt[4]{2n}\). The product rule for radicals states that \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\) when the indices are the same.
2. Apply the product rule by multiplying the expressions inside the radicals: multiply \$7m^2\( by \)2n$ to get \(7m^2 \times 2n\).
3. Combine the multiplication inside the radical: \(7m^2 \times 2n = 14m^2n\).
4. Rewrite the product as a single fourth root: \(\sqrt[4]{14m^2n}\).
5. Check if any simplification is possible by looking for perfect fourth powers inside the radical, but since none are obvious here, the expression \(\sqrt[4]{14m^2n}\) is the simplified product.