Which choices are real numbers?

General
1

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

2

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

Answer: To determine which of the given expressions are real numbers, let’s evaluate each one based on the properties of real numbers and the rules regarding roots of negative numbers:

1. A. (-256)^{1 / 11}

  • When taking the root of a negative number, the root is real if and only if the root is an odd number.
  • In this case, the root is \frac{1}{11}, which is indeed an odd number since 11 is odd.
  • Therefore, (-256)^{1/11} is a real number.

2. B. (-531441)^{1 / 12}

  • The root is \frac{1}{12} here.
  • Since 12 is an even number, taking an even root of a negative number results in an imaginary number.
  • Thus, (-531441)^{1/12} is not a real number.

3. C. (-5776)^{1 / 7}

  • The root is \frac{1}{7}.
  • Since 7 is an odd number, taking an odd root of a negative number results in a real number.
  • Therefore, (-5776)^{1/7} is a real number.

4. D. (-131072)^{1 / 16}

  • The root here is \frac{1}{16}.
  • Since 16 is an even number, taking an even root of a negative number results in an imaginary number.
  • Thus, (-131072)^{1/16} is not a real number.

Conclusion

  • Based on the properties of roots and negative numbers, the expressions that are real numbers are:
    • A. (-256)^{1/11}
    • C. (-5776)^{1/7}

Therefore, the choices A and C are real numbers.