Absolutely, let's break down the original expression step by step:
The given expression is:
[tex]\[ 2 \sqrt{27} + \sqrt{12} - 3 \sqrt{3} - 2 \sqrt{12} \][/tex]
First, let's simplify each square root term individually:
1. \( \sqrt{27} \) can be simplified as:
[tex]\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3}
\][/tex]
Therefore,
[tex]\[
2 \sqrt{27} = 2 \cdot 3 \sqrt{3} = 6 \sqrt{3}
\][/tex]
2. \( \sqrt{12} \) can be simplified as:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}
\][/tex]
3. Therefore:
[tex]\[
2 \sqrt{12} = 2 \cdot 2 \sqrt{3} = 4 \sqrt{3}
\][/tex]
Substituting these simplified terms back into the original expression, we get:
[tex]\[
6 \sqrt{3} + 2 \sqrt{3} - 3 \sqrt{3} - 4 \sqrt{3}
\][/tex]
Now, let's combine like terms:
[tex]\[
6 \sqrt{3} + 2 \sqrt{3} - 3 \sqrt{3} - 4 \sqrt{3}
\][/tex]
Combine coefficients of \( \sqrt{3} \):
[tex]\[
(6 + 2 - 3 - 4) \sqrt{3} = 1 \sqrt{3}
\][/tex]
So, the simplified form of the expression is:
[tex]\[
\sqrt{3}
\][/tex]
Thus, the correct choice is:
[tex]\[
\sqrt{3}
\][/tex]
