Let’s evaluate each option to determine which one is equivalent to [tex]\(3 \sqrt{6}\)[/tex].
### Reference Value
First, recall the expression we want to match:
[tex]\[ 3 \sqrt{6} \][/tex]
### Option A: [tex]\(\sqrt{27} \cdot \sqrt{4}\)[/tex]
We begin by simplifying the components within the square roots:
[tex]\[ \sqrt{27} = \sqrt{3^3} = 3\sqrt{3} \][/tex]
[tex]\[ \sqrt{4} = 2 \][/tex]
Hence,
[tex]\[ \sqrt{27} \cdot \sqrt{4} = (3\sqrt{3}) \cdot 2 = 6\sqrt{3} \][/tex]
### Option B: [tex]\(\sqrt{9} \cdot \sqrt{6}\)[/tex]
We simplify the square roots:
[tex]\[ \sqrt{9} = 3 \][/tex]
Thus,
[tex]\[ \sqrt{9} \cdot \sqrt{6} = 3 \cdot \sqrt{6} = 3\sqrt{6} \][/tex]
### Option C: 54
This is a simple numeric value:
[tex]\[ 54 \][/tex]
### Option D: [tex]\(\sqrt{54}\)[/tex]
We simplify the square root:
[tex]\[ \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \cdot \sqrt{6} = 3\sqrt{6} \][/tex]
### Option E: [tex]\(\sqrt{18}\)[/tex]
We simplify the square root:
[tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2} = 3\sqrt{2} \][/tex]
### Option F: [tex]\(\sqrt{27} \cdot \sqrt{2}\)[/tex]
We simplify the components within the square roots:
[tex]\[ \sqrt{27} = 3 \sqrt{3} \][/tex]
Hence,
[tex]\[ \sqrt{27} \cdot \sqrt{2} = (3\sqrt{3}) \cdot \sqrt{2} = 3 \sqrt{6} \][/tex]
Based on the calculated simplifications, we have:
- Option B and Option D both simplify directly into [tex]\(3\sqrt{6}\)[/tex].
Therefore, the choices equivalent to [tex]\(3 \sqrt{6}\)[/tex] are:
[tex]\[ \boxed{B} \][/tex]
