To determine which of the given options is a non-real complex number, let’s analyze each option one by one:
Option A: [tex]\( 5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}} \)[/tex]
- [tex]\( 5 \sqrt{\frac{1}{3}} \)[/tex] involves the square root of a positive fraction, which is a real number.
- [tex]\( \frac{9}{\sqrt{7}} \)[/tex] is also a real number because both 9 and [tex]\( \sqrt{7} \)[/tex] are real numbers.
- Subtracting two real numbers will still yield a real number.
Therefore, option A represents a real number.
Option B: [tex]\( \frac{2 + 3 \sqrt{5}}{2} \)[/tex]
- Both 2 and [tex]\( 3 \sqrt{5} \)[/tex] are real numbers.
- Dividing a real number (or combination of real numbers) by another real number (in this case, 2) results in a real number.
Therefore, option B is also a real number.
Option C: [tex]\( \frac{8}{3} + \sqrt{-\frac{7}{3}} \)[/tex]
- [tex]\( \frac{8}{3} \)[/tex] is clearly a real number.
- [tex]\( \sqrt{-\frac{7}{3}} \)[/tex] involves the square root of a negative number, which by definition is not a real number but rather a complex number.
Thus, option C includes a non-real complex number.
Option D: [tex]\( 2 - \frac{1}{\sqrt{11}} \)[/tex]
- Both 2 and [tex]\( \frac{1}{\sqrt{11}} \)[/tex] are real numbers.
- Subtracting one real number from another will produce a real number.
Therefore, option D also represents a real number.
Conclusion:
The only option that contains a non-real complex number is option C. Hence, the correct answer is:
C. [tex]\( \frac{8}{3} + \sqrt{-\frac{7}{3}} \)[/tex]
