Questions: Use the product rule to simplify the following expression. Assume that variables represent nonnegative real numbers. √(2x) · √(14x) (Type an exact answer using radicals as needed.)

Solution Steps

To simplify the expression \(\sqrt{2x} \cdot \sqrt{14x}\), we can use the product rule for square roots, which states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Therefore, we multiply the expressions under the square roots and then simplify the resulting expression.

Using the product rule for square roots, we have: \[ \sqrt{2x} \cdot \sqrt{14x} = \sqrt{(2x) \cdot (14x)} = \sqrt{28x^2} \]

Next, we simplify \(\sqrt{28x^2}\): \[ \sqrt{28x^2} = \sqrt{28} \cdot \sqrt{x^2} = \sqrt{28} \cdot x \]

The value of \(\sqrt{28}\) can be simplified as follows: \[ \sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7} \] Thus, we have: \[ \sqrt{28x^2} = 2\sqrt{7} \cdot x \]

Final Answer