QQuestionMathematics
QuestionMathematics
Which choices are real numbers? Check all that apply. A. $(- 16)^{1 / 4}$
B. $(- 4)^{1 / 2}$
C. $(- 32)^{1 / 5}$
D. $(- 8)^{1 / 3}$
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Answer
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Step 1:Let's solve this step by step:
Step 2:: Understand the definition of real numbers
- Real numbers include positive and negative rational and irrational numbers - For even roots (1 / 2, 1 / 4, etc.), the base must be non-negative - For odd roots (1 / 3, 1 / 5, etc.), the base can be negative
Step 3:: Analyze option A: (- 16)^{1 / 4}
- 1 / 4 is an even root - Base is negative - Even roots of negative numbers are NOT real numbers - Therefore, A is NOT a real number
Step 4:: Analyze option B: (- 4)^{1 / 2}
- 1 / 2 is an even root - Base is negative - Even roots of negative numbers are NOT real numbers - Therefore, B is NOT a real number
Step 5:: Analyze option C: (- 32)^{1 / 5}
- 1 / 5 is an odd root - Base is negative - Odd roots of negative numbers ARE real numbers - Therefore, C IS a real number
Step 6:: Analyze option D: (- 8)^{1 / 3}
- 1 / 3 is an odd root - Base is negative - Odd roots of negative numbers ARE real numbers - Therefore, D IS a real number
Final Answer
The real number choices are C and D.
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