1. **Stating the problem:** We want to prove that if $a,b \notin \mathbb{Q}$ (i.e., $a$ and $b$ are irrational numbers), then the product $ab$ is also irrational, i.e., $ab \notin \mathbb{Q}$. 2. **Important note:** This statement is generally **false**. The product of two irrational numbers can be rational or irrational depending on the numbers chosen. 3. **Counterexample to disprove the statement:** Consider $a = \sqrt{2}$ and $b = \sqrt{2}$. - Both $a$ and $b$ are irrational. - Their product is: $$ab = \sqrt{2} \times \sqrt{2} = 2,$$ which is rational. 4. **Conclusion:** The product of two irrational numbers is not necessarily irrational. Therefore, the statement $a,b \notin \mathbb{Q} \Rightarrow ab \notin \mathbb{Q}$ is false. If you want, we can explore conditions when the product of irrationals is irrational or rational.