Question

Use the properties of complex numbers to simplify (9+−4−−−√)+(−9−−16−−−−√) .(1 point) Responses 18-2i 18-2i 6i 6i 18−−20−−−−√ 18 minus square root of negative 20 -2i

Answer 1

To simplify the expression \( (9 + \sqrt{-4}) + \left(-\sqrt{9} - \sqrt{16}\right) \), we first rewrite the square roots of negative numbers in terms of imaginary units.

1. The term \( \sqrt{-4} \) can be rewritten as \( 2i \) because:
\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2 \cdot i = 2i.
\]

2. Next, the term \( -\sqrt{9} \) and \( -\sqrt{16} \) simplify as follows:
\[
-\sqrt{9} = -3,
\]
\[
-\sqrt{16} = -4.
\]

Thus, we can rewrite the entire expression step by step:
\[
(9 + 2i) + (-3 - 4).
\]

Now, we compute the real parts:
\[
9 - 3 - 4 = 2.
\]

Now, combining the imaginary parts:
\[
2i.
\]

Putting it all together, we have:
\[
2 + 2i.
\]

Since none of the options seem to directly match this result, I'll recheck:

1. The initial expression \( (9 + \sqrt{-4}) + (-\sqrt{9} - \sqrt{16}) \).
2. Breaking it down:
- From the first part \( 9 + \sqrt{-4} \): gives \( 9 + 2i\).
- From the second part \(-\sqrt{9} - \sqrt{16}\): gives \(-3 - 4 = -7\).

Combining all together:
\[
9 + 2i - 7 = 2 + 2i.
\]

So, unfortunately, this does not match any listed response. If the options seem to have been misstated or misunderstood, could you clarify?

For the calculations done, the answer is \( 2 + 2i \).