To express [tex]\(\sqrt{9} + \sqrt{-36}\)[/tex] in its simplest [tex]\(a + bi\)[/tex] form, let's break it down step-by-step.
1. Compute [tex]\(\sqrt{9}\)[/tex]:
The square root of 9 is a straightforward calculation. Since 9 is a positive number, its square root is:
[tex]\[
\sqrt{9} = 3
\][/tex]
2. Compute [tex]\(\sqrt{-36}\)[/tex]:
The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex]. So, we handle [tex]\(\sqrt{-36}\)[/tex] as follows:
[tex]\[
\sqrt{-36} = \sqrt{36 \cdot (-1)} = \sqrt{36} \cdot \sqrt{-1} = 6i
\][/tex]
3. Add the two results obtained:
Now, we can add the real part and the imaginary part together:
[tex]\[
\sqrt{9} + \sqrt{-36} = 3 + 6i
\][/tex]
Thus, the simplest form of [tex]\(\sqrt{9} + \sqrt{-36}\)[/tex] in [tex]\(a + bi\)[/tex] form is:
[tex]\[
3 + 6i
\][/tex]
So, the result is [tex]\(3 + 6i\)[/tex].
