To determine which choices are real numbers, we examine the expression \((-x)^{1/n}\) for each option, where \(x\) is a positive integer and \(n\) is a positive rational number.
For \( (-x)^{1/n} \) to be a real number:
1. If \(n\) is an even number, \((-x)^{1/n}\) is not a real number because the root of a negative number is not real when the root is even.
2. If \(n\) is an odd number, \((-x)^{1/n}\) is a real number because the root of a negative number is real when the root is odd.
Let's evaluate each of the given options:
Choice A: \((-531441)^{1/12}\)
- Here, \(-531441\) is the base.
- The exponent is \( \frac{1}{12} \), which translates to the twelfth root of -531441.
- Since 12 is an even number, \( (-531441)^{1/12} \) is not a real number.
Choice B: \((-131072)^{1/17}\)
- Here, \(-131072\) is the base.
- The exponent is \( \frac{1}{17} \), which translates to the seventeenth root of -131072.
- Since 17 is an odd number, \( (-131072)^{1/17} \) is a real number.
Choice C: \((-1024)^{1/5}\)
- Here, \(-1024\) is the base.
- The exponent is \( \frac{1}{5} \), which translates to the fifth root of -1024.
- Since 5 is an odd number, \( (-1024)^{1/5} \) is a real number.
Choice D: \((-256)^{1/8}\)
- Here, \(-256\) is the base.
- The exponent is \( \frac{1}{8} \), which translates to the eighth root of -256.
- Since 8 is an even number, \( (-256)^{1/8} \) is not a real number.
Based on these evaluations, the real numbers correspond to the following choices:
- B. \( (-131072)^{1/17} \)
- C. \( (-1024)^{1/5} \)
Thus, the choices that are real numbers are: [tex]\(B\)[/tex] and [tex]\(C\)[/tex].
