Simplify \(3\sqrt{45}-12\sqrt{5}+16\sqrt{20}\)leaving your answer in surd form
Answer Details
First, we need to simplify each term under the square root sign. We can write:
\begin{align*}
3\sqrt{45} &= 3\sqrt{9\times 5} = 3\times 3\sqrt{5} = 9\sqrt{5}\\
12\sqrt{5} &= 2\times 2\times 3\sqrt{5} = 4\sqrt{5}\times 3\\
16\sqrt{20} &= 16\sqrt{4\times 5} = 16\times 2\sqrt{5} = 32\sqrt{5}
\end{align*}
Now we can substitute these simplified expressions back into the original expression and simplify:
\begin{align*}
3\sqrt{45}-12\sqrt{5}+16\sqrt{20} &= 9\sqrt{5}-4\sqrt{5}\times 3+32\sqrt{5}\\
&= (9-12+32)\sqrt{5}\\
&= 29\sqrt{5}
\end{align*}
Therefore, the answer is \(29\sqrt{5}\).