Which choices are real numbers? Check all that apply.
Options
- A(−1024)^(1/4)
- B(−131072)^(1/16)
- C(−256)^(1/9)
- D(−531441)^(1/13)
Answer
The choices that are real numbers are C and D: (−256)^(1/9) and (−531441)^(1/13) respectively, as both involve odd roots of negative numbers resulting in real values.
Explanation
To determine which of the choices are real numbers, we need to evaluate the fourth root and other roots of negative numbers. **Choice A: (−1024)^(1/4) The fourth root of −1024 involves dealing with a negative number which will yield a complex number. **Choice B: (−131072)^(1/16) Similarly, the sixteenth root of −131072 will also give a complex number since we are taking an even root of a negative number. **Choice C: (−256)^(1/9) The ninth root of −256 is a real number. The exponent 1/9 is an odd root and odd roots of negative numbers are real. Evaluating this gives us −4. **Choice D: (−531441)^(1/13) The thirteenth root of −531441 is also a real number for the same reason as choice C. This evaluates to −3. Conclusion The only real numbers from the provided choices are options C and D. Thus: The correct answers are C and D.