Use the properties of complex numbers to simplify

Question

(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:

\[ \boxed{-2i} \]