To determine which of the given choices is equivalent to the product [tex]\(\sqrt{6x} \cdot \sqrt{x+3}\)[/tex], consider the simplification process.
Firstly, recognize that the product of square roots can be simplified using the property [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. In this case:
[tex]\[
\sqrt{6x} \cdot \sqrt{x+3} = \sqrt{(6x) \cdot (x+3)}
\][/tex]
Next, distribute the [tex]\(6x\)[/tex] inside the square root:
[tex]\[
(6x) \cdot (x+3) = 6x \cdot x + 6x \cdot 3
\][/tex]
Simplify the expression within the square root:
[tex]\[
6x \cdot x + 6x \cdot 3 = 6x^2 + 18x
\][/tex]
Now we have:
[tex]\[
\sqrt{6x} \cdot \sqrt{x+3} = \sqrt{6x^2 + 18x}
\][/tex]
Thus, the expression [tex]\(\sqrt{6x} \cdot \sqrt{x+3}\)[/tex] simplifies to [tex]\(\sqrt{6x^2 + 18x}\)[/tex]. Therefore, the choice equivalent to this expression is:
A. [tex]\(\sqrt{6x^2 + 18x}\)[/tex]
