Perform the operations and simplify:
$3 \sqrt{12} - 5 \sqrt{27} + 2 \sqrt{75}$
To simplify the expression \(3\sqrt{12} - 5\sqrt{27} + 2\sqrt{75}\), we begin by breaking down each radical into its simplest form. We look for perfect square factors in the radicands: \(12 = 4 \times 3\), \(27 = 9 \times 3\), and \(75 = 25 \times 3\). Using the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can rewrite each square root.
Now, we simplify each term individually:
Substitute these simplified terms back into the original expression:
\(6\sqrt{3} - 15\sqrt{3} + 10\sqrt{3}\).
Since all terms are multiples of \(\sqrt{3}\), we combine the coefficients:
\((6 - 15 + 10)\sqrt{3} = 1\sqrt{3} = \sqrt{3}\).
Comparing our result \(\sqrt{3}\) with the given options, we identify that \(\sqrt{3}\) matches one of the provided choices. The final simplified expression is \(\boxed{\sqrt{3}}\).
Simplifying square roots is a fundamental skill used in architecture and construction, such as when calculating the length of diagonal supports or the dimensions of a triangular roof section using the Pythagorean theorem. For instance, if you need to calculate a side length of \(3\sqrt{12} - 5\sqrt{27} + 2\sqrt{75}\) meters to determine the amount of material needed for a frame, simplifying the expression to \(\sqrt{3} \approx 1.732\) meters provides a much clearer understanding of the actual length required for cutting the materials.