Question
Use the properties of complex numbers to simplify (9+sqrt(-4))+(-9-sqrt(-16)). (1 point) 18-sqrt(-20) 18-2i 6i -2i
Answers
Answered by
GPT-4o mini
To simplify the expression \( (9 + \sqrt{-4}) + (-9 - \sqrt{-16}) \), we first rewrite the square roots of the negative numbers using imaginary units.
1. Calculate \( \sqrt{-4} \):
\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i
\]
2. Calculate \( \sqrt{-16} \):
\[
\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i
\]
Now substitute these values back into the expression:
\[
(9 + 2i) + (-9 - 4i)
\]
Next, combine the real and imaginary parts:
- Real parts: \( 9 + (-9) = 0 \)
- Imaginary parts: \( 2i + (-4i) = 2i - 4i = -2i \)
Thus, the simplified expression is:
\[
0 - 2i = -2i
\]
Therefore, the simplified result is \( \boxed{-2i} \).
1. Calculate \( \sqrt{-4} \):
\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i
\]
2. Calculate \( \sqrt{-16} \):
\[
\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i
\]
Now substitute these values back into the expression:
\[
(9 + 2i) + (-9 - 4i)
\]
Next, combine the real and imaginary parts:
- Real parts: \( 9 + (-9) = 0 \)
- Imaginary parts: \( 2i + (-4i) = 2i - 4i = -2i \)
Thus, the simplified expression is:
\[
0 - 2i = -2i
\]
Therefore, the simplified result is \( \boxed{-2i} \).