To determine the complex conjugate of [tex]\(8 - \sqrt{3}\)[/tex], let's follow the general rule for finding the complex conjugate.
Given a complex number in the form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers and [tex]\(i\)[/tex] is the imaginary unit, the complex conjugate is [tex]\(a - bi\)[/tex].
1. Identify the components:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -\sqrt{3}\)[/tex] (since [tex]\(8 - \sqrt{3}\)[/tex] can be seen as [tex]\(8 + (-\sqrt{3})\)[/tex]).
2. Apply the complex conjugate rule:
- The complex conjugate is [tex]\(a - bi\)[/tex]. For the given number, this means replacing [tex]\(-\sqrt{3}\)[/tex] with [tex]\(+\sqrt{3}\)[/tex].
Thus, the complex conjugate of [tex]\(8 - \sqrt{3}\)[/tex] is [tex]\[8 + \sqrt{3}\][/tex].
Checking the given options:
- [tex]\(8 \sqrt{3}\)[/tex]: Incorrect, it doesn’t follow the conjugate rule.
- [tex]\(3 + \sqrt{8}\)[/tex]: Incorrect, completely unrelated to the given number.
- [tex]\(8 + \sqrt{3}\)[/tex]: Correct, matches our derived conjugate.
- [tex]\(8 - \sqrt{3}\)[/tex]: Incorrect, this is the original number, not its conjugate.
So, the correct answer is:
[tex]\[8 + \sqrt{3}\][/tex]
The numerical result for [tex]\(8 + \sqrt{3}\)[/tex] approximately equals 9.732050807568877, confirming the correct option.
