At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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].
### 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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
