Sure, let's simplify the given expression [tex]\( \sqrt{9} + \sqrt{-36} \)[/tex] and express it in the form [tex]\( a + bi \)[/tex].
Firstly, let's tackle each term separately:
1. Simplifying [tex]\( \sqrt{9} \)[/tex]:
[tex]\[
\sqrt{9} = 3
\][/tex]
2. Simplifying [tex]\( \sqrt{-36} \)[/tex]:
[tex]\[
\sqrt{-36} = \sqrt{36 \cdot (-1)} = \sqrt{36} \cdot \sqrt{-1}
\][/tex]
We know that [tex]\( \sqrt{36} = 6 \)[/tex] and [tex]\( \sqrt{-1} = i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit.
Thus,
[tex]\[
\sqrt{-36} = 6i
\][/tex]
Now, let's combine the simplified terms:
[tex]\[
\sqrt{9} + \sqrt{-36} = 3 + 6i
\][/tex]
So, the expression [tex]\( \sqrt{9} + \sqrt{-36} \)[/tex] simplified in the form [tex]\( a + bi \)[/tex] is [tex]\( 3 + 6i \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{3 + 6i}
\][/tex]
