To simplify the expression \((9 + \sqrt{-4}) + (-9 - \sqrt{-16})\), we need to express the square roots of the negative numbers in terms of complex numbers.
First, we can express \(\sqrt{-4}\) and \(\sqrt{-16}\) as follows:
\[ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \]
\[ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \]
Now we can substitute these values into the original expression:
\[ (9 + \sqrt{-4}) + (-9 - \sqrt{-16}) = (9 + 2i) + (-9 - 4i) \]
Next, we can combine the real and imaginary parts:
For the real part: \[ 9 - 9 = 0 \]
For the imaginary part: \[ 2i - 4i = -2i \]
Putting it all together, we have:
\[ 0 - 2i = -2i \]
So, the simplified expression is:
\(-2i\)