To determine which of the given values are real numbers, we need to analyze the nature of the roots involved when negative bases are raised to fractional exponents.
We have four different choices to examine:
### Choice A:
[tex]\[
A. (-131072)^{1 / 17}
\][/tex]
Since [tex]\(-131072\)[/tex] is a negative number, raising it to the [tex]\( \frac{1}{17}\)[/tex] power involves taking the 17th root. The 17th root of a negative number is complex because the 17th root does not yield a real number for negative bases.
### Choice B:
[tex]\[
B. (-531441)^{1 / 12}
\][/tex]
Similarly, [tex]\(-531441\)[/tex] is negative. Raising it to the [tex]\( \frac{1}{12}\)[/tex] power implies taking the 12th root. The 12th root of a negative number produces a complex number since the 12th root of a negative value does not result in a real number.
### Choice C:
[tex]\[
C. (-256)^{1 / 8}
\][/tex]
[tex]\(-256\)[/tex] is negative, and raising it to the [tex]\( \frac{1}{8}\)[/tex] power means finding the 8th root. The 8th root of a negative number is also complex, as negative bases raised to even roots do not produce real results.
### Choice D:
[tex]\[
D. (-1024)^{1 / 5}
\][/tex]
For [tex]\(-1024\)[/tex], raising it to the [tex]\( \frac{1}{5}\)[/tex] power involves taking the 5th root. The 5th root of a negative number is also complex. Negative bases raised to odd roots still end up being complex numbers when considered in the context of fractional exponents.
### Conclusion:
None of the given choices result in real numbers. This is because raising a negative number to any fractional power (that involves taking roots) results in a complex number.
Therefore, the analysis reveals that:
[tex]\[
\boxed{\text{None of the above choices are real numbers.}}
\][/tex]
The real number status for each of the choices A, B, C, and D is:
- [tex]\(A:\, \text{Not real} \)[/tex]
- [tex]\(B:\, \text{Not real} \)[/tex]
- [tex]\(C:\, \text{Not real} \)[/tex]
- [tex]\(D:\, \text{Not real} \)[/tex]
Thus, the result is:
[tex]\[
(False, False, False, False)
\][/tex]
