Which choices are real numbers? Check all that apply.
A. (-16)^1/4
B. (-4)^1/2
C. (-32)^1/5
D. (-8)^1/3
The Correct Answer and Explanation is :
The correct answer is: C and D
To determine which choices are real numbers, we must evaluate the expressions using the rules of exponents and how they apply to negative bases.
Option A: ((-16)^{1/4})
This represents the fourth root of (-16). In the set of real numbers, we cannot take an even root (like a square root or fourth root) of a negative number because it does not yield a real number. The result of ((-16)^{1/4}) would be complex.
Thus, Option A is not a real number.
Option B: ((-4)^{1/2})
This represents the square root of (-4). Like in Option A, taking an even root (in this case, a square root) of a negative number results in an imaginary number, not a real number. The square root of (-4) is (2i), where (i) is the imaginary unit.
Thus, Option B is not a real number.
Option C: ((-32)^{1/5})
This represents the fifth root of (-32). Odd roots (like the cube root or fifth root) of negative numbers are allowed in the real number system because the result is a real number. For instance, the fifth root of (-32) is (-2), since ((-2)^5 = -32).
Thus, Option C is a real number.
Option D: ((-8)^{1/3})
This represents the cube root of (-8). Like in Option C, the cube root of a negative number results in a real number. The cube root of (-8) is (-2), since ((-2)^3 = -8).
Thus, Option D is a real number.
Conclusion:
The real numbers are C and D, because both involve taking odd roots of negative numbers, which yield real results.