Given the expression [tex]\(6 \sqrt{3}\)[/tex], let's verify which of the given choices are equivalent to it.
Choice A: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
First, we simplify [tex]\(\sqrt{36}\)[/tex]:
[tex]\[
\sqrt{36} = 6
\][/tex]
So,
[tex]\[
\sqrt{3} \cdot \sqrt{36} = \sqrt{3} \cdot 6 = 6 \sqrt{3}
\][/tex]
This matches the given expression. Therefore, Choice A is equivalent.
Choice B: [tex]\(\sqrt{108}\)[/tex]
Let's simplify [tex]\(\sqrt{108}\)[/tex]:
[tex]\[
108 = 36 \cdot 3
\][/tex]
So,
[tex]\[
\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \cdot \sqrt{3} = 6 \sqrt{3}
\][/tex]
This also matches the given expression. Therefore, Choice B is equivalent.
Choice C: 108
The given expression is [tex]\(6 \sqrt{3}\)[/tex], which is clearly not equal to 108, as 108 is a scalar whereas [tex]\(6 \sqrt{3}\)[/tex] is an irrational number. Therefore, Choice C is not equivalent.
Choice D: [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]
Let's simplify [tex]\(\sqrt{18} \cdot \sqrt{6}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2}
\][/tex]
[tex]\[
\sqrt{6} = \sqrt{2 \cdot 3} = \sqrt{2} \cdot \sqrt{3}
\][/tex]
So,
[tex]\[
\sqrt{18} \cdot \sqrt{6} = (3 \cdot \sqrt{2}) \cdot (\sqrt{2} \cdot \sqrt{3}) = 3 \cdot 2 \cdot \sqrt{3} = 6 \sqrt{3}
\][/tex]
This matches the given expression. Therefore, Choice D is equivalent.
Choice E: [tex]\(\sqrt{54}\)[/tex]
Let's simplify [tex]\(\sqrt{54}\)[/tex]:
[tex]\[
54 = 9 \cdot 6
\][/tex]
So,
[tex]\[
\sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \cdot \sqrt{6}
\][/tex]
This does not match the given expression [tex]\(6 \sqrt{3}\)[/tex]. Therefore, Choice E is not equivalent.
Choice F: [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]
Simplify [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]:
[tex]\[
\sqrt{3} \cdot \sqrt{6} = \sqrt{18}
\][/tex]
And we have already simplified [tex]\(\sqrt{18}\)[/tex] in Choice D and found it to be [tex]\(3 \sqrt{2}\)[/tex], which does not match [tex]\(6 \sqrt{3}\)[/tex]. Therefore, Choice F is not equivalent.
In summary, the choices that are equivalent to [tex]\(6 \sqrt{3}\)[/tex] are:
- Choice A: [tex]\(\sqrt{3} \cdot \sqrt{36}\)[/tex]
- Choice B: [tex]\(\sqrt{108}\)[/tex]
