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

Question

and 23
.

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

To analyze who is correct, let's take a closer look at the numbers involved:

1. The number \(6 - \sqrt{6}\).
2. The number \(23\).

• \(6 - \sqrt{6}\) is a mixture of a rational number (6) and an irrational number (\(\sqrt{6}\)). Therefore, \(6 - \sqrt{6}\) is irrational because it includes an irrational component.
• The number \(23\) is rational as it is an integer.

Now we need to determine the product of these two numbers: \[ (6 - \sqrt{6}) \times 23 \] An important property of real numbers is that the product of a rational number and an irrational number is always irrational. Since \(23\) is rational and \(6 - \sqrt{6}\) is irrational, their product must be irrational.

Now let's evaluate the statements made by Elaina, Miguel, and Swati:

• Elaina: Says the answer is irrational because both numbers are irrational — Incorrect (only one number is irrational).
• Miguel: Says the answer is rational because both numbers are rational — Incorrect (one number is rational, and one is irrational).
• Swati: Says the answer is irrational because one number is rational and the other number is irrational — Correct.

Therefore, Swati is the one who is correct.

The correct answer is: Swati