Let's analyze each of the given expressions step by step to determine which one results in a non-real complex number.
### Expression 1:
[tex]\[
\frac{4}{\sqrt{3}} - 7 \sqrt{\frac{3}{5}}
\][/tex]
- The term [tex]\(\frac{4}{\sqrt{3}}\)[/tex] is a real number since [tex]\(\sqrt{3}\)[/tex] is a real number and dividing real numbers results in a real number.
- The term [tex]\(\sqrt{\frac{3}{5}}\)[/tex] is also a real number because the square root of a positive number is real.
- Thus, the entire expression is a subtraction of two real numbers, resulting in a real number.
### Expression 2:
[tex]\[
\frac{5}{2} - \sqrt{15}
\][/tex]
- The term [tex]\(\frac{5}{2}\)[/tex] is a real number.
- The term [tex]\(\sqrt{15}\)[/tex] is a real number because [tex]\(\sqrt{15}\)[/tex] is the square root of a positive number.
- Hence, the expression is a subtraction of two real numbers, resulting in a real number.
### Expression 3:
[tex]\[
\frac{2}{3} + \sqrt{-\frac{27}{28}}
\][/tex]
- The term [tex]\(\frac{2}{3}\)[/tex] is a real number.
- The term [tex]\(\sqrt{-\frac{27}{28}}\)[/tex] involves the square root of a negative number.
- [tex]\(\frac{27}{28}\)[/tex] is a positive fraction, but the negative sign inside the square root makes it a complex number.
- The square root of a negative number is an imaginary number, so [tex]\(\sqrt{-\frac{27}{28}} = i \sqrt{\frac{27}{28}}\)[/tex].
- Therefore, this expression adds a real number and an imaginary number, resulting in a complex number.
### Expression 4:
[tex]\[
\frac{-4 + \sqrt{7}}{3}
\][/tex]
- The term [tex]\(\sqrt{7}\)[/tex] is a real number since [tex]\(\sqrt{7}\)[/tex] is the square root of a positive number.
- The numerator [tex]\(-4 + \sqrt{7}\)[/tex] is a real number because it is the sum of two real numbers.
- Dividing a real number by another real number (3) also results in a real number.
### Conclusion:
Among the given expressions, Expression 3:
[tex]\[
\frac{2}{3} + \sqrt{-\frac{27}{28}}
\][/tex]
is the one that results in a non-real complex number.
