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Which of these expressions represents the product of an irrational number and a rational number
being irrational?

To determine which expression represents the product of an irrational number and a rational number, we need to examine each option:

1. \( \sqrt{3} \cdot \sqrt{49} \)
   - \( \sqrt{3} \) is irrational.
   - \( \sqrt{49} = 7 \), which is rational.
   - Product: \( \sqrt{3} \cdot 7 \), which is irrational.

2. \( 0 \cdot \sqrt{4} \)
   - \( 0 \) is rational.
   - \( \sqrt{4} = 2 \), which is rational.
   - Product: \( 0 \), which is rational (and specifically equal to 0).

3. \( \sqrt{2} \cdot e \)
   - \( \sqrt{2} \) is irrational.
   - \( e \) (Euler's number) is irrational.
   - Product: \( \sqrt{2} \cdot e \), which is irrational, but this isn't the product of an irrational and a rational number.

4. \( \pi \cdot \sqrt{7} \)
   - \( \pi \) is irrational.
   - \( \sqrt{7} \) is irrational.
   - Product: \( \pi \cdot \sqrt{7} \), which is irrational, but again, this isn't the product of an irrational and a rational number.

The correct choice based on the criteria "the product of an irrational number and a rational number being irrational" is the first option:

**Option 1: \( \sqrt{3} \cdot \sqrt{49} \)**

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How do we define a rational number and an irrational number?
2. Why does the product of a rational and an irrational number result in an irrational number?
3. Can the product of two irrational numbers ever be rational?
4. What are some other examples of irrational numbers?
5. How do square roots of non-perfect squares differ from square roots of perfect squares?

**Tip:** Remember that the product of a non-zero rational number with an irrational number is always irrational.

Explore how to determine the product of an irrational number and a rational number being irrational. This mathematical analysis covers examples like sqrt(3) * sqrt(49) and discusses why such products result in an irrational number. Suitable for Grades 9-12.

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