Sure, let's simplify the expression step by step.
We are given:
[tex]\[ 3 \sqrt{3} \cdot 6 \sqrt{6} \][/tex]
First, we will multiply the coefficients (the numbers outside the square roots):
[tex]\[ 3 \cdot 6 = 18 \][/tex]
Next, we use the property of square roots that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. Applying this to [tex]\(\sqrt{3} \cdot \sqrt{6}\)[/tex]:
[tex]\[ \sqrt{3} \cdot \sqrt{6} = \sqrt{3 \cdot 6} = \sqrt{18} \][/tex]
Now we need to simplify [tex]\(\sqrt{18}\)[/tex]:
[tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \][/tex]
So, substituting back, we have:
[tex]\[ 18 \cdot \sqrt{18} = 18 \cdot 3 \sqrt{2} = 54 \sqrt{2} \][/tex]
Thus, the simplified form of the expression [tex]\( 3 \sqrt{3} \cdot 6 \sqrt{6} \)[/tex] is:
[tex]\[ 54 \sqrt{2} \][/tex]
Therefore, the correct answer is:
A. [tex]\(54 \sqrt{2}\)[/tex]
