What is the following difference?
$11 \sqrt{45}-4 \sqrt{5}$
The expression to simplify is \(11\sqrt{45} - 4\sqrt{5}\). To perform this subtraction, we first need to simplify the square root term \(\sqrt{45}\) to see if it shares a common radical part with \(4\sqrt{5}\). We know that \(45 = 9 \times 5\). Using the property of radicals that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we can rewrite \(\sqrt{45}\) as \(\sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}\).
Now substitute this simplified form back into our original expression:
\[11\sqrt{45} - 4\sqrt{5} = 11(3\sqrt{5}) - 4\sqrt{5}\]
Performing the multiplication \(11 \times 3\sqrt{5}\), we get \(33\sqrt{5}\). Now the expression is:
\[33\sqrt{5} - 4\sqrt{5}\]
Since both terms now share the same radical \(\sqrt{5}\), we can subtract the coefficients directly by factoring out the \(\sqrt{5}\):
\[(33 - 4)\sqrt{5} = 29\sqrt{5}\]
Comparing this result to the given options, we find that the correct choice is \(29\sqrt{5}\).
Radical simplification is widely used in fields like architecture and engineering. For example, if you are designing a square garden where the side length is \(\sqrt{45}\) meters and you subtract a section that is \(\sqrt{5}\) meters long, simplifying these terms is essential for accurate material estimation. Mastering these algebraic manipulations helps in efficiently solving real-world measurement problems.