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Which choices are real numbers? Check all that apply.
A. [tex](-531441)^{1 / 12}[/tex] B. [tex](-131072)^{1 / 16}[/tex] C. [tex](-5776)^{1 / 7}[/tex] D. [tex](-256)^{1 / 11}[/tex]
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.
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