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Which of the following is a rational number: \(√(32)\), 81, 90, or \(√(125)\)?

A. \(√(32)\)
B. 81
C. 90
D. \(√(125)\)

3 answers

To determine which of the given numbers is a rational number, we need to examine each option.

  1. \(√(32)\): \[ √(32) = √(16 \cdot 2) = √16 \cdot √2 = 4√2. \] Since \(√2\) is an irrational number, \(4√2\) is also irrational. Therefore, \(√(32)\) is not a rational number.

  2. 81: The number 81 is a whole number, which means it can be expressed as the fraction \(\frac{81}{1}\). Hence, 81 is a rational number.

  3. 90: Similar to 81, the number 90 can also be expressed as the fraction \(\frac{90}{1}\). Therefore, 90 is a rational number.

  4. \(√(125)\): \[ √(125) = √(25 \cdot 5) = √25 \cdot √5 = 5√5. \] Since \(√5\) is an irrational number, \(5√5\) is also irrational. Thus, \(√(125)\) is not a rational number.

The rational numbers from the options provided are 81 and 90.

So the answer choices are:

  • A. \(√(32)\) - Not rational.
  • B. 81 - Rational.
  • C. 90 - Rational.
  • D. \(√(125)\) - Not rational.

Thus, both B (81) and C (90) are rational numbers, but the question likely asks for one. If we have to select one:

The answer is 81.

So, the final answer is B. 81.

wrong

I apologize for any confusion. Let's clarify the rationality of each option again.

  1. \(√(32)\): \[ √(32) = √(16 \cdot 2) = √16 \cdot √2 = 4√2. \] Since \(√2\) is irrational, \(√(32)\) is also irrational.

  2. 81: 81 is a whole number and can be expressed as the fraction \(\frac{81}{1}\), which makes it a rational number.

  3. 90: 90 is also a whole number and can be expressed as the fraction \(\frac{90}{1}\), so it is a rational number as well.

  4. \(√(125)\): \[ √(125) = √(25 \cdot 5) = √25 \cdot √5 = 5√5. \] Since \(√5\) is irrational, \(√(125)\) is also irrational.

From the analysis:

  • \(√(32)\) is not rational.
  • 81 is rational.
  • 90 is rational.
  • \(√(125)\) is not rational.

The numbers that are rational are 81 and 90. Both options B (81) and C (90) are rational numbers. Since the question asked for a single rational number among the options, either would be correct, but if we focus on 81 first strictly:

The correct answer is B. 81.

If there was ambiguity due to selection or preference indicated, please let me know, and I can help clarify further!