Question

question
fully simplify.
\\((9\sqrt{6})(2\sqrt{-30})\\)
answer attempt 1 out of 2

Explanation:

Step1: Recall the property of imaginary numbers

We know that $\sqrt{-a}=i\sqrt{a}$ for $a>0$. So, $\sqrt{-30}=i\sqrt{30}$.
The expression becomes $(9\sqrt{6})(2\times i\sqrt{30})$.

Step2: Multiply the coefficients and the radicals separately

First, multiply the coefficients: $9\times2 = 18$.
Then, multiply the radicals: $\sqrt{6}\times\sqrt{30}=\sqrt{6\times30}=\sqrt{180}$.

Step3: Simplify the radical $\sqrt{180}$

We can factor $180$ as $36\times5$, so $\sqrt{180}=\sqrt{36\times5}=\sqrt{36}\times\sqrt{5}=6\sqrt{5}$.

Step4: Combine the results

Now, we have $18\times i\times6\sqrt{5}$.
Multiply $18$ and $6$: $18\times6 = 108$.
So the simplified form is $108i\sqrt{5}$.

Answer:

$108i\sqrt{5}$