To determine [tex]\(\sqrt{-16} + \sqrt{49}\)[/tex] in the form [tex]\(a + b i\)[/tex], we need to evaluate each term separately and then combine the results.
First, let’s find [tex]\(\sqrt{-16}\)[/tex]:
Since [tex]\(-16\)[/tex] is a negative number, its square root will involve an imaginary component. Recall that [tex]\(\sqrt{-1} = i\)[/tex].
[tex]\[
\sqrt{-16} = \sqrt{16 \cdot (-1)} = \sqrt{16} \cdot \sqrt{-1} = 4i
\][/tex]
Next, let's find [tex]\(\sqrt{49}\)[/tex]:
[tex]\[
\sqrt{49} = 7
\][/tex]
Now, combine these results:
[tex]\[
\sqrt{-16} + \sqrt{49} = 4i + 7
\][/tex]
To express this in the standard complex number form [tex]\(a + bi\)[/tex], the real part [tex]\(a\)[/tex] is 7 and the imaginary part [tex]\(b\)[/tex] is 4.
Therefore, the final expression is:
[tex]\[
7 + 4i
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{7 + 4i}
\][/tex]
This corresponds to option C.
