Simplify the following expression by combining like radicals.
$6 \sqrt{54}-\sqrt{600}$
The objective is to simplify the expression \(6\sqrt{54} - \sqrt{600}\). To do this, we need to express each radical in its simplest form. A radical \(\sqrt{n}\) is in its simplest form when the radicand \(n\) has no perfect square factors other than 1. If a radical can be written as \(k\sqrt{m}\), then two terms can be combined if they share the same radical part \(\sqrt{m}\).
First, let us simplify \(\sqrt{54}\). We look for perfect square factors of 54. Since \(54 = 9 \times 6 = 3^2 \times 6\), we can write:
\[\sqrt{54} = \sqrt{3^2 \times 6} = 3\sqrt{6}\]
Now, substitute this back into the first part of the expression:
\[6\sqrt{54} = 6 \times (3\sqrt{6}) = 18\sqrt{6}\]
Next, we simplify \(\sqrt{600}\). We look for perfect square factors of 600. Since \(600 = 100 \times 6 = 10^2 \times 6\), we can write:
\[\sqrt{600} = \sqrt{10^2 \times 6} = 10\sqrt{6}\]
Now that both terms are expressed in terms of \(\sqrt{6}\), we can combine them:
\[6\sqrt{54} - \sqrt{600} = 18\sqrt{6} - 10\sqrt{6}\]
By factoring out \(\sqrt{6}\), we get:
\[(18 - 10)\sqrt{6} = 8\sqrt{6}\]
The expression simplifies to \(\boxed{8\sqrt{6}}\).
Simplifying radicals is a fundamental skill used in architecture and construction when dealing with diagonal measurements. For example, if you are calculating the length of a diagonal brace in a frame that measures \(\sqrt{54}\) meters, and you need to subtract a portion of length \(\sqrt{600}/6\) meters, you would use this method to find the exact remaining length needed for the brace. Mastering this ensures precision in building materials and design.