To simplify the expression [tex]\(2 \sqrt{18} + 3 \sqrt{2} + \sqrt{162}\)[/tex], we need to break down each term into a simpler form. Here's the step-by-step process:
1. Simplify [tex]\(2 \sqrt{18}\)[/tex]:
- First, we express 18 as a product of a square number and another number: [tex]\(18 = 9 \times 2\)[/tex].
- So, [tex]\( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2} \)[/tex].
- Now, multiplying by 2: [tex]\( 2 \sqrt{18} = 2 \times 3 \sqrt{2} = 6 \sqrt{2} \)[/tex].
2. Simplify [tex]\(3 \sqrt{2}\)[/tex]:
- This term is already in its simplest form.
3. Simplify [tex]\(\sqrt{162}\)[/tex]:
- First, we express 162 as a product of a square number and another number: [tex]\(162 = 81 \times 2\)[/tex].
- So, [tex]\( \sqrt{162} = \sqrt{81 \times 2} = \sqrt{81} \times \sqrt{2} = 9 \sqrt{2} \)[/tex].
Now, we can combine all the simplified terms:
[tex]\[
2 \sqrt{18} + 3 \sqrt{2} + \sqrt{162} = 6 \sqrt{2} + 3 \sqrt{2} + 9 \sqrt{2}.
\][/tex]
Next, we 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 [tex]\(2 \sqrt{18} + 3 \sqrt{2} + \sqrt{162}\)[/tex] is [tex]\( \boxed{18 \sqrt{2}} \)[/tex].
