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 index \(n\) is the same for both radicals.
2. Apply the product rule by combining the expressions inside the radicals: \(\sqrt[4]{7m^2} \cdot \sqrt[4]{2n} = \sqrt[4]{(7m^2)(2n)}\).
3. Multiply the terms inside the radical: \$7 \times 2 = 14\(, so the expression inside becomes \)14m^2n$.
4. Rewrite the product under a single fourth root: \(\sqrt[4]{14m^2n}\).
5. Confirm that the expression cannot be simplified further by checking if any factors inside the radical are perfect fourth powers.