Certainly! Let's analyze which expressions are equivalent to the given expression [tex]\( 3 \sqrt{6} \)[/tex].
First, let's start by looking at each option one by one.
### Option A: [tex]\(\sqrt{27} \cdot \sqrt{4}\)[/tex]
1. Simplify [tex]\(\sqrt{27}\)[/tex]:
[tex]\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3}
\][/tex]
2. Simplify [tex]\(\sqrt{4}\)[/tex]:
[tex]\[
\sqrt{4} = 2
\][/tex]
3. Combine them:
[tex]\[
\sqrt{27} \cdot \sqrt{4} = 3 \sqrt{3} \cdot 2 = 6 \sqrt{3}
\][/tex]
Comparing [tex]\( 3 \sqrt{6} \)[/tex] with [tex]\( 6 \sqrt{3} \)[/tex], these are not equivalent.
### Option B: [tex]\(\sqrt{27} \cdot \sqrt{2}\)[/tex]
1. Simplify:
[tex]\[
\sqrt{27} \cdot \sqrt{2} = \sqrt{27 \cdot 2} = \sqrt{54}
\][/tex]
We will compare [tex]\(\sqrt{54}\)[/tex] in Option C to the original expression [tex]\(3 \sqrt{6}\)[/tex] later, but let's proceed as though it simplifies correctly.
### Option C: [tex]\(\sqrt{54}\)[/tex]
1. Simplify:
[tex]\[
\sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6}
\][/tex]
Thus, [tex]\(\sqrt{54}\)[/tex] is indeed equivalent to [tex]\(3 \sqrt{6}\)[/tex].
### Option D: [tex]\(\sqrt{18}\)[/tex]
1. Simplify:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2}
\][/tex]
Comparing [tex]\(3 \sqrt{6}\)[/tex] with [tex]\(3 \sqrt{2}\)[/tex], these are not equivalent.
### Option E: 54
1. This is just a number, not involving any square roots.
[tex]\[
54 \neq 3 \sqrt{6}
\][/tex]
It is clearly not equivalent.
### Option F: [tex]\(\sqrt{9} \cdot \sqrt{6}\)[/tex]
1. Simplify:
[tex]\[
\sqrt{9} \cdot \sqrt{6} = 3 \cdot \sqrt{6} = 3 \sqrt{6}
\][/tex]
This matches the original expression exactly.
### Conclusion
The expressions that are equivalent to [tex]\(3 \sqrt{6}\)[/tex] are:
- B. [tex]\(\sqrt{27} \cdot \sqrt{2}\)[/tex]
- C. [tex]\(\sqrt{54}\)[/tex]
- F. [tex]\(\sqrt{9} \cdot \sqrt{6}\)[/tex]
Thus, the correct choices are B, C, and F.
