Sure, let's solve the given expression step-by-step:
We want to multiply [tex]\(\sqrt{3x}\)[/tex] with [tex]\((\sqrt{42x} + 3\sqrt{3})\)[/tex]:
[tex]\[
\sqrt{3x} \left( \sqrt{42x} + 3\sqrt{3} \right)
\][/tex]
We use the distributive property to expand this:
[tex]\[
(\sqrt{3x}) (\sqrt{42x}) + (\sqrt{3x}) (3\sqrt{3})
\][/tex]
First, let's simplify each term individually.
1. Simplifying [tex]\(\sqrt{3x} \cdot \sqrt{42x}\)[/tex]:
[tex]\[
\sqrt{3x} \cdot \sqrt{42x} = \sqrt{(3x) \cdot (42x)} = \sqrt{126x^2}
\][/tex]
Since [tex]\(x^2\)[/tex] is a perfect square, [tex]\(\sqrt{126x^2}\)[/tex] becomes:
[tex]\[
\sqrt{126} \cdot x = \sqrt{126}x = \sqrt{(3 \cdot 42)}x = \sqrt{(3 \cdot 6 \cdot 7)}x = \sqrt{(3 \cdot 3 \cdot 2 \cdot 7)}x = 3\sqrt{14}x
\][/tex]
2. Simplifying [tex]\(\sqrt{3x} \cdot 3\sqrt{3}\)[/tex]:
[tex]\[
\sqrt{3x} \cdot 3\sqrt{3} = 3\sqrt{3} \cdot \sqrt{3x} = 3 (\sqrt{3} \cdot \sqrt{3} \cdot \sqrt{x}) = 3 \cdot 3 \cdot \sqrt{x} = 9\sqrt{x}
\][/tex]
So, combining these two results, we get:
[tex]\[
3\sqrt{14}x + 9\sqrt{x}
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\sqrt{3x} \left( \sqrt{42x} + 3\sqrt{3} \right)\)[/tex] is:
[tex]\[
3\sqrt{14}x + 9\sqrt{x}
\][/tex]
