To simplify the expression [tex]\(2 \sqrt{18} + 3 \sqrt{2} + \sqrt{162}\)[/tex], we can follow these steps:
1. Simplify [tex]\(\sqrt{18}\)[/tex]: Rewrite 18 as a product of its prime factors:
[tex]\[
18 = 9 \times 2 = 3^2 \times 2
\][/tex]
So, [tex]\(\sqrt{18} = \sqrt{3^2 \times 2} = 3 \sqrt{2}\)[/tex].
2. Simplify [tex]\(2 \sqrt{18}\)[/tex]:
Since [tex]\(\sqrt{18} = 3 \sqrt{2}\)[/tex],
[tex]\[
2 \sqrt{18} = 2 \times 3 \sqrt{2} = 6 \sqrt{2}
\][/tex]
3. Simplify [tex]\(\sqrt{162}\)[/tex]: Rewrite 162 as a product of its prime factors:
[tex]\[
162 = 81 \times 2 = 9^2 \times 2
\][/tex]
So, [tex]\(\sqrt{162} = \sqrt{9^2 \times 2} = 9 \sqrt{2}\)[/tex].
4. Combine all terms:
Now that we have simplified each term individually, we combine them:
[tex]\[
2 \sqrt{18} + 3 \sqrt{2} + \sqrt{162} = 6 \sqrt{2} + 3 \sqrt{2} + 9 \sqrt{2}
\][/tex]
Add the coefficients of [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
6 \sqrt{2} + 3 \sqrt{2} + 9 \sqrt{2} = (6 + 3 + 9) \sqrt{2} = 18 \sqrt{2}
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
18 \sqrt{2}
\][/tex]
So, the correct answer is [tex]\(\boxed{18 \sqrt{2}}\)[/tex].
