To simplify the expression [tex]\( 3 \sqrt{108} + 2 \sqrt{75} - \sqrt{48} \)[/tex], let's go through each term step-by-step.
Step 1: Simplify [tex]\(\sqrt{108}\)[/tex]
First, we'll simplify [tex]\(\sqrt{108}\)[/tex]:
[tex]\[
\sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}
\][/tex]
Therefore,
[tex]\[
3 \sqrt{108} = 3 \times 6\sqrt{3} = 18\sqrt{3}
\][/tex]
Step 2: Simplify [tex]\(\sqrt{75}\)[/tex]
Next, let's simplify [tex]\(\sqrt{75}\)[/tex]:
[tex]\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
\][/tex]
Therefore,
[tex]\[
2 \sqrt{75} = 2 \times 5\sqrt{3} = 10\sqrt{3}
\][/tex]
Step 3: Simplify [tex]\(\sqrt{48}\)[/tex]
Finally, we'll simplify [tex]\(\sqrt{48}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}
\][/tex]
Therefore,
[tex]\[
-\sqrt{48} = -4\sqrt{3}
\][/tex]
Step 4: Combine All Terms
Now we add up all the simplified terms:
[tex]\[
3 \sqrt{108} + 2 \sqrt{75} - \sqrt{48} = 18\sqrt{3} + 10\sqrt{3} - 4\sqrt{3}
\][/tex]
Combine the coefficients of [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
18\sqrt{3} + 10\sqrt{3} - 4\sqrt{3} = (18 + 10 - 4)\sqrt{3} = 24\sqrt{3}
\][/tex]
Thus, the simplified form of the expression [tex]\( 3 \sqrt{108} + 2 \sqrt{75} - \sqrt{48} \)[/tex] is:
[tex]\[
\boxed{24\sqrt{3}}
\][/tex]
