Question

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

sqaureroot 15x times squareroot135x

sqaureroot15x times sqaureroot135x = blank (Type an exact answer using radicals as needed.)

Answer 1

To simplify the expression \(\sqrt{15x} \times \sqrt{135x}\), we can use the product rule for square roots, which states that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

So, we have:

\[
\sqrt{15x} \times \sqrt{135x} = \sqrt{(15x) \times (135x)} = \sqrt{15 \times 135 \times x^2}
\]

Now we can calculate \(15 \times 135\):

\[
15 \times 135 = 2025
\]

Now substituting that back into the radical:

\[
\sqrt{15 \times 135 \times x^2} = \sqrt{2025 \times x^2}
\]

Next, we can simplify \(\sqrt{2025}\) and \(\sqrt{x^2}\):

The square root of \(x^2\) is simply \(x\) (as \(x\) is a nonnegative real number). Now we need to simplify \(\sqrt{2025}\). The prime factorization of \(2025\) is:

\[
2025 = 5^2 \times 3^4
\]

Thus, we can take the square root:

\[
\sqrt{2025} = \sqrt{5^2} \times \sqrt{3^4} = 5 \times 9 = 45
\]

Putting it all together, we get:

\[
\sqrt{2025 \times x^2} = 45x
\]

Therefore,

\[
\sqrt{15x} \times \sqrt{135x} = 45x
\]

The final simplified expression is:

\[
\boxed{45x}
\]