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:
\[ \boxed{45x} \]
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