To determine which of these options is a non-real complex number, let's analyze each option step-by-step, focusing on whether the result contains an imaginary component.
### Option A: [tex]\(\frac{9 + 3 \sqrt{5}}{2}\)[/tex]
- The expression inside the numerator is [tex]\(9 + 3 \sqrt{5}\)[/tex].
- Here, [tex]\(\sqrt{5}\)[/tex] is a real number, so [tex]\(3 \sqrt{5}\)[/tex] is also real.
- Adding [tex]\(9\)[/tex] (a real number) to [tex]\(3 \sqrt{5}\)[/tex] results in a real number.
- Dividing this real number by [tex]\(2\)[/tex] will also yield a real number.
Thus, [tex]\(\frac{9 + 3 \sqrt{5}}{2}\)[/tex] is a real number.
### Option B: [tex]\(\frac{8}{3} + \sqrt{-\frac{7}{3}}\)[/tex]
- The expression [tex]\(\frac{8}{3}\)[/tex] is a real number.
- However, [tex]\(\sqrt{-\frac{7}{3}}\)[/tex] involves taking the square root of a negative number.
- The square root of a negative number is an imaginary number.
Thus, [tex]\(\frac{8}{3} + \sqrt{-\frac{7}{3}}\)[/tex] is a non-real complex number because it contains an imaginary part.
### Option C: [tex]\(2 - \frac{1}{\sqrt{11}}\)[/tex]
- The expression [tex]\(2\)[/tex] is a real number.
- The term [tex]\(\sqrt{11}\)[/tex] is a real number, so [tex]\(\frac{1}{\sqrt{11}}\)[/tex] is also real.
- Subtracting one real number from another real number results in a real number.
Thus, [tex]\(2 - \frac{1}{\sqrt{11}}\)[/tex] is a real number.
### Option D: [tex]\(5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}}\)[/tex]
- The expression [tex]\(\sqrt{\frac{1}{3}}\)[/tex] is a real number, so [tex]\(5 \sqrt{\frac{1}{3}}\)[/tex] is also real.
- The term [tex]\(\sqrt{7}\)[/tex] is a real number, so [tex]\(\frac{9}{\sqrt{7}}\)[/tex] is also real.
- Subtracting one real number from another real number results in a real number.
Thus, [tex]\(5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}}\)[/tex] is a real number.
### Conclusion
Among the given options, the only one that is a non-real complex number is option B: [tex]\(\frac{8}{3} + \sqrt{-\frac{7}{3}}\)[/tex].
