Question

jacob says that the product of a rational number and an irrational number is sometimes rational and sometimes irrational.

which statement correctly describes whether jacob is correct or incorrect?

• jacob is correct; the product of a rational number and an irrational number may be either rational or irrational.
• jacob is incorrect; the product of a rational number and an irrational number is always rational.
• jacob is incorrect; the product of a rational number and an irrational number is always irrational.

Explanation:

Step1: Analyze the product with zero

If the rational number is \(0\) and the irrational number is \(\sqrt{2}\):

$$0 \times \sqrt{2} = 0$$

Since \(0\) is a rational number, the product is rational.

Step2: Analyze the product with non-zero

If the rational number is \(2\) and the irrational number is \(\sqrt{2}\):

$$2 \times \sqrt{2} = 2\sqrt{2}$$

Since \(2\sqrt{2}\) cannot be expressed as a fraction of integers, the product is irrational.

Step3: Determine the correct statement

The product can be either rational or irrational, so Jacob is correct.

Answer:

Jacob is correct; the product of a rational number and an irrational number may be either rational or irrational.