Question

Which of the following numbers will be a rational number?

• A. $\dfrac{\sqrt{3}}{\sqrt{5}}$
• B. $\sqrt{2} \times \sqrt{7}$
• C. $(\sqrt{5} + \sqrt{7})(\sqrt{5} - \sqrt{7})$
• D. $\sqrt{12}$

Hint:

Always check if the expression simplifies using $(a+b)(a-b)=a^2-b^2$. Such products often turn into rational numbers.

Answer 1

The Correct Option is C

Step 1: Simplify each option
- Option (A): $\dfrac{\sqrt{3}}{\sqrt{5}} = \sqrt{\dfrac{3}{5}}$ which is irrational.
- Option (B): $\sqrt{2} \times \sqrt{7} = \sqrt{14}$ which is irrational.
- Option (C): $(\sqrt{5} + \sqrt{7})(\sqrt{5} - \sqrt{7})$ \[ = (\sqrt{5})^2 - (\sqrt{7})^2 = 5 - 7 = -2 \] This is rational.
- Option (D): $\sqrt{12} = 2\sqrt{3}$ which is irrational.

Step 2: Conclusion
The only rational value comes from option (C).
The correct answer is option (C).