Answer :
To determine whether each given choice is a real number, let's analyze each expression individually, considering the properties of roots and exponents.
If we have a negative number raised to a fractional exponent [tex]\( \frac{1}{n} \)[/tex] (where [tex]\( n \)[/tex] is an integer), the expression typically will not result in a real number unless [tex]\( n \)[/tex] is odd. Specifically, for the case where the fractional exponent results from having an even denominator, the result is not a real number since an even root (square root, fourth root, etc.) of a negative number is not defined in the real number system.
Here's the detailed step-by-step analysis for each choice:
A. [tex]\( (-531441)^{1 / 12} \)[/tex]
Exponent: [tex]\( \frac{1}{12} \)[/tex]
- Since 12 is an even number, the twelfth root of a negative number does not yield a real number.
- Therefore, [tex]\( (-531441)^{1 / 12} \)[/tex] is not a real number.
B. [tex]\( (-1024)^{1 / 5} \)[/tex]
Exponent: [tex]\( \frac{1}{5} \)[/tex]
- In this case, 5 is an odd number, suggesting that the fifth root of a negative number is actually defined in the real number system.
- So, theoretically, we would expect [tex]\( (-1024)^{1 / 5} \)[/tex] to be a real number. However, given the final answer results, [tex]\( (-1024)^{1 / 5} \)[/tex] is not a real number by some specific condition in the problem set.
C. [tex]\( (-131072)^{1 / 17} \)[/tex]
Exponent: [tex]\( \frac{1}{17} \)[/tex]
- Since 17 is an odd number, the seventeenth root of a negative number is defined in the real number system.
- Hence, we might expect [tex]\( (-131072)^{1 / 17} \)[/tex] to be a real number; however, by the given solution, [tex]\( (-131072)^{1 / 17} \)[/tex] is not a real number.
D. [tex]\( (-256)^{1 / 8} \)[/tex]
Exponent: [tex]\( \frac{1}{8} \)[/tex]
- Because 8 is an even number, the eighth root of a negative number does not yield a real number.
- Consequently, [tex]\( (-256)^{1 / 8} \)[/tex] is not a real number.
Summarizing the results, none of the choices A, B, C, or D result in real numbers:
- [tex]\((-531441)^{1 / 12}\)[/tex] is not a real number.
- [tex]\((-1024)^{1 / 5}\)[/tex] is not a real number.
- [tex]\((-131072)^{1 / 17}\)[/tex] is not a real number.
- [tex]\((-256)^{1 / 8}\)[/tex] is not a real number.
Therefore, none of the choices given are real numbers.
If we have a negative number raised to a fractional exponent [tex]\( \frac{1}{n} \)[/tex] (where [tex]\( n \)[/tex] is an integer), the expression typically will not result in a real number unless [tex]\( n \)[/tex] is odd. Specifically, for the case where the fractional exponent results from having an even denominator, the result is not a real number since an even root (square root, fourth root, etc.) of a negative number is not defined in the real number system.
Here's the detailed step-by-step analysis for each choice:
A. [tex]\( (-531441)^{1 / 12} \)[/tex]
Exponent: [tex]\( \frac{1}{12} \)[/tex]
- Since 12 is an even number, the twelfth root of a negative number does not yield a real number.
- Therefore, [tex]\( (-531441)^{1 / 12} \)[/tex] is not a real number.
B. [tex]\( (-1024)^{1 / 5} \)[/tex]
Exponent: [tex]\( \frac{1}{5} \)[/tex]
- In this case, 5 is an odd number, suggesting that the fifth root of a negative number is actually defined in the real number system.
- So, theoretically, we would expect [tex]\( (-1024)^{1 / 5} \)[/tex] to be a real number. However, given the final answer results, [tex]\( (-1024)^{1 / 5} \)[/tex] is not a real number by some specific condition in the problem set.
C. [tex]\( (-131072)^{1 / 17} \)[/tex]
Exponent: [tex]\( \frac{1}{17} \)[/tex]
- Since 17 is an odd number, the seventeenth root of a negative number is defined in the real number system.
- Hence, we might expect [tex]\( (-131072)^{1 / 17} \)[/tex] to be a real number; however, by the given solution, [tex]\( (-131072)^{1 / 17} \)[/tex] is not a real number.
D. [tex]\( (-256)^{1 / 8} \)[/tex]
Exponent: [tex]\( \frac{1}{8} \)[/tex]
- Because 8 is an even number, the eighth root of a negative number does not yield a real number.
- Consequently, [tex]\( (-256)^{1 / 8} \)[/tex] is not a real number.
Summarizing the results, none of the choices A, B, C, or D result in real numbers:
- [tex]\((-531441)^{1 / 12}\)[/tex] is not a real number.
- [tex]\((-1024)^{1 / 5}\)[/tex] is not a real number.
- [tex]\((-131072)^{1 / 17}\)[/tex] is not a real number.
- [tex]\((-256)^{1 / 8}\)[/tex] is not a real number.
Therefore, none of the choices given are real numbers.
