Question

use the product rule to simplify the following expression. assume that variables represent nonnegative real numbers.\\(sqrt{5x} cdot sqrt{15x}\\)\\(sqrt{5x} cdot sqrt{15x} = square\\)\\((\text{type an exact answer using radicals as needed.})\\)

Explanation:

Step1: Apply product rule for radicals

$\sqrt{5x} \cdot \sqrt{15x} = \sqrt{5x \cdot 15x}$

Step2: Multiply terms inside radical

$\sqrt{5 \cdot 15 \cdot x \cdot x} = \sqrt{75x^2}$

Step3: Factor into perfect squares

$\sqrt{25 \cdot 3 \cdot x^2} = \sqrt{25x^2 \cdot 3}$

Step4: Simplify perfect square radicals

$\sqrt{25x^2} \cdot \sqrt{3} = 5x\sqrt{3}$

Answer:

$5x\sqrt{3}$