Prove or give a counterexample: If x and y are irrational, then xy is irrational.

The statement **"If \( x \) and \( y \) are irrational, then \( xy \) is irrational"** is **false**. A counterexample can demonstrate that this is not always true.

### Counterexample:

Let \( x = \sqrt{2} \) and \( y = \frac{1}{\sqrt{2}} \).

- Both \( x \) and \( y \) are irrational numbers. Specifically, \( \sqrt{2} \) is known to be irrational.
- However, \( xy = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1 \), and **1 is a rational number**.

Thus, we have two irrational numbers \( x \) and \( y \) whose product is a rational number, proving that the original statement is false.

### Conclusion:
It is not always true that the product of two irrational numbers is irrational. There are cases, like the one shown above, where the product is rational.

Would you like further details or have any questions?

Here are five related questions to expand on this concept:
1. Can the product of two irrational numbers ever be irrational?
2. Are there any conditions under which the product of two irrational numbers is guaranteed to be irrational?
3. What are some other examples where the product of irrational numbers is rational?
4. Is the sum of two irrational numbers always irrational?
5. How can you determine whether the product of two specific numbers is rational or irrational?

**Tip:** When working with irrational numbers, remember that their behavior under operations like multiplication and addition is not always straightforward—counterexamples often provide powerful insights.

Explore whether the product of two irrational numbers is always irrational with a counterexample using √2 and 1/√2. Understand the properties of irrational numbers and learn why their product can sometimes be rational. Suitable for Grades 10-12.

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