Elaina, Miguel, and Swati are calculating the product of 6–√ and 2/3

Question

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Elaina says the answer is irrational because both numbers are irrational.

Miguel says the answer is rational because both numbers are rational.

Swati says the answer is irrational because one number is rational and the other number is irrational.

Who is correct? Enter the number that corresponds to the correct answer.

Elaina
Miguel
Swati

Answer 1

Let's analyze the numbers involved in the calculation. The expression is \(6 - \sqrt{6}\) and \(\frac{2}{3}\).

1. Identifying the nature of the numbers:

   • \(6\) is a rational number.
   • \(\sqrt{6}\) is an irrational number.
   • Therefore, \(6 - \sqrt{6}\) combines a rational number \(6\) with an irrational number \(\sqrt{6}\), resulting in an irrational number.
   • \(\frac{2}{3}\) is a rational number.

2. Now let's formally define the product:

   • The product is \((6 - \sqrt{6}) \cdot \frac{2}{3}\).

3. Rational and Irrational products:

   • The product of a rational number and an irrational number is always irrational.

Given this analysis, here is the summary of the statements made:

• Elaina: Incorrect because she claims both numbers are irrational. Only \(6 - \sqrt{6}\) is irrational, while \(\frac{2}{3}\) is rational.
• Miguel: Incorrect because he claims both numbers are rational. One number is irrational.
• Swati: Correct because she states that the product of a rational number (\(\frac{2}{3}\)) and an irrational number (\(6 - \sqrt{6}\)) will indeed be irrational.

Thus, the correct answer is Swati.