Simplify: $-\sqrt{5}-3 \sqrt{36}+6 \sqrt{5}$
The given expression is \(-\sqrt{5} - 3\sqrt{36} + 6\sqrt{5}\). To simplify this, we need to address two parts: evaluating the perfect square root and combining the terms that share the same radical component, \(\sqrt{5}\).
First, identify the term containing a perfect square. We know that \(6^2 = 6 \times 6 = 36\). Therefore, the square root of 36 is 6. Substituting this into our expression gives:
\[-\sqrt{5} - 3(6) + 6\sqrt{5}\]
Performing the multiplication \(3 \times 6 = 18\), we get:
\[-\sqrt{5} - 18 + 6\sqrt{5}\]
Now, we can group the terms that involve \(\sqrt{5}\). The expression is now \((6\sqrt{5} - \sqrt{5}) - 18\). Factoring out the \(\sqrt{5}\) from the first two terms:
\[(6 - 1)\sqrt{5} - 18\]
\[5\sqrt{5} - 18\]
This matches the simplified form required.
Comparing our result \(5\sqrt{5} - 18\) with the given options, it is clear that the correct choice is \(5\sqrt{5} - 18\). The final simplified value is \(\boxed{5\sqrt{5} - 18}\).
Simplifying radicals is a fundamental skill in algebra, commonly used in fields such as engineering and physics. For example, when calculating the side length of a triangle using the Pythagorean theorem or working with vectors, you often end up with sums of square roots. Simplifying these expressions allows for easier comparisons and more accurate measurements, just as we simplified our expression here to its most compact form.