To find the equivalent expression for the product [tex]\(\sqrt{6 x^2} \cdot \sqrt{18 x^2}\)[/tex], follow these steps:
1. Use the property of square roots:
We know that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. Therefore,
[tex]\[
\sqrt{6 x^2} \cdot \sqrt{18 x^2} = \sqrt{(6 x^2) \cdot (18 x^2)}.
\][/tex]
2. Calculate the product inside the square root:
Multiply the terms inside the square root:
[tex]\[
6 x^2 \cdot 18 x^2 = 108 x^4.
\][/tex]
3. Simplify the expression:
Now, we have [tex]\(\sqrt{108 x^4}\)[/tex]. We can simplify this further by factoring 108 as [tex]\(36 \cdot 3\)[/tex]:
[tex]\[
\sqrt{108 x^4} = \sqrt{36 \cdot 3 \cdot x^4}.
\][/tex]
4. Separate the square root for easier computation:
Recognize that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Thus,
[tex]\[
\sqrt{36 \cdot 3 \cdot x^4} = \sqrt{36} \cdot \sqrt{3} \cdot \sqrt{x^4}.
\][/tex]
5. Simplify each square root:
[tex]\(\sqrt{36} = 6\)[/tex] and [tex]\(\sqrt{x^4} = x^2\)[/tex] (since [tex]\(x \geq 0\)[/tex], we can directly take the square root):
[tex]\[
\sqrt{36} \cdot \sqrt{3} \cdot \sqrt{x^4} = 6 \cdot \sqrt{3} \cdot x^2.
\][/tex]
6. Combine the simplified terms:
Therefore, the expression simplifies to:
[tex]\[
6 x^2 \sqrt{3}.
\][/tex]
By following the steps, we determine that the equivalent expression is [tex]\(6 x^2 \sqrt{3}\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{6 x^2 \sqrt{3}}. \][/tex]
