Question

complex numbers online practice
complete this assessment to review what youve learned. it will not count toward your grade.
use the properties of complex numbers to simplify $(8 + \sqrt{-16}) + (5 - \sqrt{-36})$ (1 point)
\\(\circ \sqrt{-8} + \sqrt{41}\\)
\\(\circ 13 + 10i\\)
\\(\circ 13 - 2i\\)
\\(\circ 13 + \sqrt{-52}\\)

Explanation:

Step1: Simplify square roots of negative numbers

Recall that \( \sqrt{-a} = i\sqrt{a} \) for \( a > 0 \). So, \( \sqrt{-16}=i\sqrt{16}=4i \) and \( \sqrt{-36}=i\sqrt{36}=6i \).
The expression becomes \( (8 + 4i)+(5 - 6i) \).

Step2: Combine like terms

Combine the real parts: \( 8 + 5 = 13 \).
Combine the imaginary parts: \( 4i-6i=-2i \).
So the simplified form is \( 13 - 2i \)? Wait, no, wait. Wait, original imaginary parts: \( +4i - 6i = -2i \)? Wait, no, the second term is \( - \sqrt{-36} \), so \( -6i \). So \( 4i-6i=-2i \)? Wait, but let's re - check. Wait, the first complex number is \( 8 + \sqrt{-16}=8 + 4i \), the second is \( 5-\sqrt{-36}=5 - 6i \). Then adding them: \( 8 + 5+(4i-6i)=13 - 2i \)? But wait, the options have \( 13 + 10i \), \( 13 - 2i \), etc. Wait, maybe I made a mistake. Wait, \( \sqrt{-16}=4i \), so \( 8 + 4i \), and \( \sqrt{-36}=6i \), so \( 5-6i \). Then \( (8 + 4i)+(5 - 6i)=8 + 5+4i-6i = 13-2i \). But wait, the option is there: \( 13 - 2i \) is an option. Wait, but let's check again. Wait, maybe I misread the signs. Wait, the problem is \( (8+\sqrt{-16})+(5 - \sqrt{-36}) \). So \( \sqrt{-16}=4i \), \( \sqrt{-36}=6i \), so \( 8 + 4i+5 - 6i=(8 + 5)+(4i-6i)=13-2i \). So the correct option is \( 13 - 2i \).

Wait, but let's check the options again. The options are:

1. \( \sqrt{-8}+\sqrt{41} \)
2. \( 13 + 10i \)
3. \( 13 - 2i \)
4. \( 13+\sqrt{-52} \)

So the correct one is \( 13 - 2i \), which is the third option.

Answer:

\( 13 - 2i \) (the option with text "13 − 2i")