Question

Which of the following numbers is 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:

Use the identity \( (a+b)(a-b)=a^2-b^2 \) to simplify radical expressions quickly. If the radicals cancel completely, the result is rational.

Answer 1

The Correct Option is C

Step 1: Recall the identity used.
\[ (a + b)(a - b) = a^2 - b^2 \]
Step 2: Apply it to option (C).
\[ (\sqrt{5} + \sqrt{7})(\sqrt{5} - \sqrt{7}) = (\sqrt{5})^2 - (\sqrt{7})^2 = 5 - 7 = -2 \]
Step 3: Identify the nature of the result.
The result is \(-2\), which is a rational number.

Step 4: Verify other options.
(A) \( \frac{\sqrt{3}}{\sqrt{5}} = \sqrt{\frac{3}{5}} \) — irrational because the square root of a non-perfect fraction is irrational.
(B) \( \sqrt{2} \times \sqrt{7} = \sqrt{14} \) — irrational.
(D) \( \sqrt{12} = 2\sqrt{3} \) — also irrational.
Step 5: Conclusion.
Hence, the only rational result is from option (C).