To determine which of the given choices are real numbers, we need to examine each expression and see if the result yields a real number.
### Choice A: [tex]\((-531441)^{1 / 12}\)[/tex]
To find [tex]\((-531441)^{1 / 12}\)[/tex]:
[tex]\[
(-531441)^{1 / 12} \approx -3.0
\][/tex]
Since [tex]\(-3.0\)[/tex] is a real number, Choice A is a real number.
### Choice B: [tex]\((-131072)^{1 / 16}\)[/tex]
To find [tex]\((-131072)^{1 / 16}\)[/tex]:
[tex]\[
(-131072)^{1 / 16} \approx -2.0885475648548275
\][/tex]
Since [tex]\(-2.0885475648548275\)[/tex] is a real number, Choice B is a real number.
### Choice C: [tex]\((-5776)^{1 / 7}\)[/tex]
To find [tex]\((-5776)^{1 / 7}\)[/tex]:
[tex]\[
(-5776)^{1 / 7} \approx -3.4464764246261175
\][/tex]
Since [tex]\(-3.4464764246261175\)[/tex] is a real number, Choice C is a real number.
### Choice D: [tex]\((-256)^{1 / 11}\)[/tex]
To find [tex]\((-256)^{1 / 11}\)[/tex]:
[tex]\[
(-256)^{1 / 11} \approx -1.6555065597696215
\][/tex]
Since [tex]\(-1.6555065597696215\)[/tex] is a real number, Choice D is a real number.
Therefore, all four choices, A, B, C, and D, are real numbers.
