Question

Give an example of two irrational numbers, whose product is an irrational number.

• A. $\sqrt{3},\sqrt{3}$
• B. $\sqrt{2},\sqrt{2}$
• C. $\sqrt{2},-\sqrt{2}$
• D. $\sqrt{2},\sqrt{3}$

Answer 1

Answer: (D)

Consider option $A$.

The number are $\sqrt{3}$ and $\sqrt{3}$.
Their product $=\sqrt{3}\times \sqrt{3}=3$, which is a rational number.

Consider option $B$.

The number are $\sqrt{2}$ and $\sqrt{2}$.
Their product $=\sqrt{2}\times \sqrt{2}=2$, which is a rational number.

Consider option $C$.

The number are $\sqrt{2}$ and $-\sqrt{2}$.
Their product $=\left(\sqrt{2}\right)\times (-\sqrt{2})=-2$, which is a rational number.

Consider option $D$.

The number are $\sqrt{2}$ and $\sqrt{3}$

Their product $=\sqrt{2}\times \left(\sqrt{3}\right)=\sqrt{6}$, which is an irrational number.

Therefore, option $D$ is correct.