To simplify the expression [tex]\(3 \sqrt{108} + 2 \sqrt{75} - \sqrt{48}\)[/tex], we first need to reduce each square root term into its simplest form by factoring out squares.
1. Simplify [tex]\( \sqrt{108} \)[/tex]:
[tex]\[ 108 = 36 \times 3 \][/tex]
[tex]\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6 \sqrt{3} \][/tex]
So, [tex]\( 3 \sqrt{108} \)[/tex] becomes:
[tex]\[ 3 \times 6 \sqrt{3} = 18 \sqrt{3} \][/tex]
2. Simplify [tex]\( \sqrt{75} \)[/tex]:
[tex]\[ 75 = 25 \times 3 \][/tex]
[tex]\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5 \sqrt{3} \][/tex]
So, [tex]\( 2 \sqrt{75} \)[/tex] becomes:
[tex]\[ 2 \times 5 \sqrt{3} = 10 \sqrt{3} \][/tex]
3. Simplify [tex]\( \sqrt{48} \)[/tex]:
[tex]\[ 48 = 16 \times 3 \][/tex]
[tex]\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \sqrt{3} \][/tex]
So, [tex]\( \sqrt{48} \)[/tex] becomes:
[tex]\[ 1 \times 4 \sqrt{3} = 4 \sqrt{3} \][/tex]
Now, substitute these simplified forms back into the original expression:
[tex]\[ 3 \sqrt{108} + 2 \sqrt{75} - \sqrt{48} = 18 \sqrt{3} + 10 \sqrt{3} - 4 \sqrt{3} \][/tex]
Combine the like terms:
[tex]\[ 18 \sqrt{3} + 10 \sqrt{3} - 4 \sqrt{3} = (18 + 10 - 4) \sqrt{3} = 24 \sqrt{3} \][/tex]
Thus, the simplified expression is:
[tex]\[ 24 \sqrt{3} \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{24 \sqrt{3}} \][/tex]
