Question

Use the product rule to simplify the following expression. Assume that variables represent nonnegative real numbers.
squareroot 5x^2 times squareroot 15x


squareroot 5x^2 times squareroot 15x (Please type an exact answer using radicals as needed.)

Answer 1

To simplify the expression \(\sqrt{5x^2} \times \sqrt{15x}\), we can use the property of square roots that states \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

1. Combine under the square root:
\[
\sqrt{5x^2} \times \sqrt{15x} = \sqrt{5x^2 \times 15x}
\]

2. Multiply the expressions inside the square root:
\[
5x^2 \times 15x = 75x^3
\]

3. Now we have:
\[
\sqrt{75x^3}
\]

4. Factor \(75\) into \(25 \times 3\) to extract the square root:
\[
\sqrt{75x^3} = \sqrt{25 \times 3 \times x^2 \times x} = \sqrt{25} \times \sqrt{3} \times \sqrt{x^2} \times \sqrt{x}
\]

5. Simplify the square roots:
- \(\sqrt{25} = 5\),
- \(\sqrt{x^2} = x\).

Thus, we have:
\[
\sqrt{75x^3} = 5 \times \sqrt{3} \times x \times \sqrt{x} = 5x\sqrt{3x}
\]

The final simplified expression is:
\[
\boxed{5x\sqrt{3x}}
\]