Use the product rule to multiply the following.
$\sqrt{5x}\cdot \sqrt{7y}$

• A. $\sqrt{35xy}$
• B. $35xy$
• C. $\sqrt{12x^2{y}^2}$
• D. $\sqrt{12xy}$

Answer: (A)

Verified step by step guidance

1. Recall the product rule for square roots: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). This means you can multiply the expressions inside the square roots together under a single square root.
2. Identify the expressions inside each square root: the first is \$5x\( and the second is \)7y$.
3. Multiply the expressions inside the square roots: \$5x \times 7y = 35xy$.
4. Write the product under a single square root using the product rule: \(\sqrt{5x} \cdot \sqrt{7y} = \sqrt{35xy}\).
5. If needed, check if the expression under the square root can be simplified further by factoring perfect squares, but in this case, \$35xy$ does not have perfect square factors that simplify the root.