Question

Use the product rule to simplify the following expression. Assume that variables represent nonnegative real numbers.

squareroot7x times squareroot 42x

squareroot 7x times squareroot 42x = blank (Please type an exact answer using radicals as needed.)

Answer 1

To simplify the expression \(\sqrt{7x} \times \sqrt{42x}\), we can use the property of square roots that states \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

Applying this property, we get:

\[
\sqrt{7x} \times \sqrt{42x} = \sqrt{(7x) \times (42x)}
\]

Now, calculate \(7 \times 42\) and \(x \times x\):

\[
7 \times 42 = 294
\]
\[
x \times x = x^2
\]

Now we combine these results:

\[
\sqrt{7x} \times \sqrt{42x} = \sqrt{294x^2}
\]

Next, we can simplify \(\sqrt{294x^2}\):

Since \(x^2\) is a perfect square, we can take it out of the square root:

\[
\sqrt{294x^2} = \sqrt{294} \times \sqrt{x^2} = \sqrt{294} \times x
\]

Therefore, the final simplified expression is:

\[
\sqrt{7x} \times \sqrt{42x} = x\sqrt{294}
\]

Thus, the answer is:

\[
x\sqrt{294}
\]