Question

express in simplest radical form.
$-10\sqrt{45} + 5\sqrt{20}$

Explanation:

Step1: Simplify each radical

First, we simplify \(\sqrt{45}\) and \(\sqrt{20}\) by factoring out perfect squares.
For \(\sqrt{45}\), we have \(45 = 9\times5\), so \(\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\) (since \(\sqrt{9} = 3\)).
For \(\sqrt{20}\), we have \(20 = 4\times5\), so \(\sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}\) (since \(\sqrt{4} = 2\)).

Step2: Substitute the simplified radicals back into the expression

Now we substitute these simplified forms back into the original expression \(-10\sqrt{45}+5\sqrt{20}\).
Substituting \(\sqrt{45}=3\sqrt{5}\) and \(\sqrt{20}=2\sqrt{5}\), we get:
\(-10\times(3\sqrt{5}) + 5\times(2\sqrt{5})\)

Step3: Perform the multiplication

Next, we perform the multiplication operations.
\(-10\times3\sqrt{5}=-30\sqrt{5}\) and \(5\times2\sqrt{5}=10\sqrt{5}\).
So the expression becomes \(-30\sqrt{5}+10\sqrt{5}\).

Step4: Combine like terms

Now we combine the like terms (terms with the same radical part \(\sqrt{5}\)).
\(-30\sqrt{5}+10\sqrt{5}=(-30 + 10)\sqrt{5}\)
Calculating \(-30 + 10\) gives \(-20\), so we have \(-20\sqrt{5}\).

Answer:

\(-20\sqrt{5}\)