To determine which of the provided options is a complex number, let us first recall that a complex number typically includes an imaginary part. The imaginary part arises when we take the square root of a negative number, denoted as [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
Now, let's analyze each option step by step:
Option A: [tex]\(7 - \sqrt{\frac{5}{4}}\)[/tex]
- Here, we have a real number 7 and a square root of a positive fraction [tex]\(\frac{5}{4}\)[/tex].
- [tex]\(\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}\)[/tex].
- Both 7 and [tex]\(\frac{\sqrt{5}}{2}\)[/tex] are real numbers.
- The result is a real number: [tex]\(7 - \frac{\sqrt{5}}{2}\)[/tex].
Option B: [tex]\(-\sqrt{11} - 8\)[/tex]
- This option involves [tex]\(-\sqrt{11}\)[/tex], which is a real number because it is the square root of a positive number.
- Subtracting 8, also a real number, does not introduce an imaginary part.
- The result is a real number: [tex]\(-\sqrt{11} - 8\)[/tex].
Option C: [tex]\(\frac{1}{2} + \sqrt{-81}\)[/tex]
- The term [tex]\(\sqrt{-81}\)[/tex] indicates that we are taking the square root of a negative number.
- This means it will have an imaginary part: [tex]\(\sqrt{-81} = 9i\)[/tex], as [tex]\(81\)[/tex] is a perfect square.
- Adding any real number, [tex]\(\frac{1}{2}\)[/tex] in this case, to an imaginary number results in a complex number.
- The result is a complex number: [tex]\(\frac{1}{2} + 9i\)[/tex].
Option D: [tex]\(\frac{\sqrt{13} + 4}{3}\)[/tex]
- Here, [tex]\(\sqrt{13}\)[/tex] is a real number because 13 is positive.
- Adding another real number, 4, and then dividing by 3 still results in a real number.
- The result is a real number: [tex]\(\frac{\sqrt{13} + 4}{3}\)[/tex].
From the evaluations above, only Option C: [tex]\(\frac{1}{2} + \sqrt{-81}\)[/tex] is identified to be a complex number because it involves the square root of a negative number, which introduces an imaginary part.
Therefore, the answer is Option C.
