Question

what is the following difference?
$11\sqrt{45} - 4\sqrt{5}$
options: $14\sqrt{10}$, $95\sqrt{5}$, $7\sqrt{40}$, $29\sqrt{5}$

Explanation:

Step1: Simplify \(\sqrt{45}\)

We know that \(45 = 9\times5\), so \(\sqrt{45}=\sqrt{9\times5}\). By the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{9\times5}=\sqrt{9}\times\sqrt{5} = 3\sqrt{5}\). Then \(11\sqrt{45}=11\times3\sqrt{5}=33\sqrt{5}\).

Step2: Subtract the two terms

Now we have \(11\sqrt{45}-4\sqrt{5}=33\sqrt{5}-4\sqrt{5}\). Since the terms have the same radical part (\(\sqrt{5}\)), we can combine them: \((33 - 4)\sqrt{5}=29\sqrt{5}\).

Answer:

\(29\sqrt{5}\) (the option with \(29\sqrt{5}\))