To find all the complex roots for a given number, we can check each set of complex roots provided and identify which set correctly corresponds to the roots of a certain complex number. The problem presents several options, and let's go through each one step-by-step.
Given the problem: Find all the complex roots of [tex]\(-8i\)[/tex].
We need to verify each provided set to determine the correct one.
### Option 1: [tex]\(2 i, -\sqrt{3}+i, -\sqrt{3}+i\)[/tex]:
1. Confirm the roots:
- [tex]\(2i\)[/tex]
- [tex]\(-\sqrt{3}+i\)[/tex]
- [tex]\(-\sqrt{3}+i\)[/tex] (Note: This root repeats, so it is unlikely a valid option)
### Option 2: [tex]\(2 i, -\sqrt{3}-i, \sqrt{3}-i\)[/tex]:
1. Confirm the roots:
- [tex]\(2i\)[/tex]
- [tex]\(-\sqrt{3}-i\)[/tex]
- [tex]\(\sqrt{3}-i\)[/tex]
### Option 3: [tex]\(-2 i, \sqrt{3}+i, \sqrt{3}-i\)[/tex]:
1. Confirm the roots:
- [tex]\(-2i\)[/tex]
- [tex]\(\sqrt{3}+i\)[/tex]
- [tex]\(\sqrt{3}-i\)[/tex]
### Option 4: [tex]\(-2 i, \sqrt{3}+i, \sqrt{3}+i\)[/tex]:
1. Confirm the roots:
- [tex]\(-2i\)[/tex]
- [tex]\(\sqrt{3}+i\)[/tex]
- [tex]\(\sqrt{3}+i\)[/tex] (Note: This root repeats, so it is unlikely a valid option)
### Option 5: [tex]\(-2 i, \sqrt{3}-i, \sqrt{3}-i\)[/tex]:
1. Confirm the roots:
- [tex]\(-2i\)[/tex]
- [tex]\(\sqrt{3}-i\)[/tex]
- [tex]\(\sqrt{3}-i\)[/tex] (Note: This root repeats, so it is unlikely a valid option)
From the above analysis, there are multiple candidate sets where the roots do not repeat and likely represent different exponents in the cube root equation.
After carefully analyzing and reasoning through the roots, we can find out which are the valid tuples representing valid roots [tex]\(2i\)[/tex], [tex]\(-\sqrt{3}-i\)[/tex], [tex]\(\sqrt{3}-i\)[/tex], [tex]\(-2i\)[/tex], [tex]\(\sqrt{3}+i\)[/tex], [tex]\(\sqrt{3}-i\)[/tex].
After cross-verifying the options, the correct tuple is:
[tex]\[
-2 i, \sqrt{3}+i, \sqrt{3}-i
\][/tex]
This set does not repeat any roots and forms the necessary complex roots for -8i.
Thus, the correct answer is:
[tex]\[
(-0-2j)
\][/tex]
