What is the simplified form of the following expression?
[tex]\[ 2 \sqrt{27} + \sqrt{12} - 3 \sqrt{3} - 2 \sqrt{12} \][/tex]

A. \(\sqrt{3}\)

B. \(9 \sqrt{3}\)

C. \(11 \sqrt{3}\)

D. [tex]\(15 \sqrt{3}\)[/tex]



Answer :

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]