Question

Use the properties of complex numbers to simplify
(9+square root of -4) plus (-9-square root of -16)

Answer 1

To simplify the expression \( (9 + \sqrt{-4}) + (-9 - \sqrt{-16}) \), we first simplify the square root terms.

1. **Simplify \(\sqrt{-4}\)**:
\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i
\]

2. **Simplify \(\sqrt{-16}\)**:
\[
\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i
\]

Now, we can replace the square root terms in the original expression:

\[
(9 + 2i) + (-9 - 4i)
\]

3. **Combine the real parts and the imaginary parts**:
- The real parts are \(9\) and \(-9\), which combine to:
\[
9 - 9 = 0
\]
- The imaginary parts are \(2i\) and \(-4i\), which combine to:
\[
2i - 4i = -2i
\]

Putting it all together, we get:

\[
0 - 2i = -2i
\]

Thus, the simplified expression is:

\[
\boxed{-2i}
\]