Let's analyze each of the given radicals step-by-step to determine which ones are real numbers and which are not:
1. [tex]\(\sqrt{5}\)[/tex]:
The square root of a positive number is a real number. Since 5 is a positive number, [tex]\(\sqrt{5}\)[/tex] is real.
2. [tex]\(-\sqrt{13}\)[/tex]:
The square root of a positive number (in this case, 13) is real. Taking the negative of a real number still yields a real number. Therefore, [tex]\(-\sqrt{13}\)[/tex] is real.
3. [tex]\(\sqrt[3]{-10}\)[/tex]:
The cube root of a number can be real even if the number is negative. Cube roots of negative numbers are real because multiplying three negative factors together results in a negative number. Hence, [tex]\(\sqrt[3]{-10}\)[/tex] is real.
4. [tex]\(\sqrt{-26}\)[/tex]:
The square root of a negative number is not a real number because there is no real number that, when squared, gives a negative result. Therefore, [tex]\(\sqrt{-26}\)[/tex] is not real.
Based on the above analysis:
- [tex]\(\sqrt{5}\)[/tex] is real.
- [tex]\(-\sqrt{13}\)[/tex] is real.
- [tex]\(\sqrt[3]{-10}\)[/tex] is real.
- [tex]\(\sqrt{-26}\)[/tex] is not real.
Therefore, the radical of [tex]\(\sqrt{-26}\)[/tex] is not a real number.
