To simplify the expression [tex]\(3 \sqrt{12} - 5 \sqrt{27} + 2 \sqrt{75}\)[/tex], we need to first break down each term by factoring out the square roots as much as possible.
1. Simplify [tex]\(3 \sqrt{12}\)[/tex]:
[tex]\[
12 = 4 \times 3
\][/tex]
So,
[tex]\[
3 \sqrt{12} = 3 \sqrt{4 \cdot 3} = 3 \cdot \sqrt{4} \cdot \sqrt{3} = 3 \cdot 2 \cdot \sqrt{3} = 6 \sqrt{3}
\][/tex]
2. Simplify [tex]\(5 \sqrt{27}\)[/tex]:
[tex]\[
27 = 9 \times 3
\][/tex]
So,
[tex]\[
5 \sqrt{27} = 5 \sqrt{9 \cdot 3} = 5 \cdot \sqrt{9} \cdot \sqrt{3} = 5 \cdot 3 \cdot \sqrt{3} = 15 \sqrt{3}
\][/tex]
3. Simplify [tex]\(2 \sqrt{75}\)[/tex]:
[tex]\[
75 = 25 \times 3
\][/tex]
So,
[tex]\[
2 \sqrt{75} = 2 \sqrt{25 \cdot 3} = 2 \cdot \sqrt{25} \cdot \sqrt{3} = 2 \cdot 5 \cdot \sqrt{3} = 10 \sqrt{3}
\][/tex]
Now that each term is simplified, we can combine them:
[tex]\[
3 \sqrt{12} - 5 \sqrt{27} + 2 \sqrt{75} = 6 \sqrt{3} - 15 \sqrt{3} + 10 \sqrt{3}
\][/tex]
Combine like terms:
[tex]\[
(6 - 15 + 10) \sqrt{3} = 1 \sqrt{3} = \sqrt{3}
\][/tex]
So, the simplified form of the original expression is:
[tex]\[
\boxed{\sqrt{3}}
\][/tex]
Based on the calculated result, the numerical value of [tex]\(\sqrt{3}\)[/tex] is approximately [tex]\(1.7320508075688772\)[/tex], and this confirms the correctness of our simplification steps.
