Let x = √3 + √5 and y = √8 + √2 . Which is greater?

Question

Let x = √3 + √5 and y = √8 + √2 . Which is greater?

1. A. x = y
2. B. x y
3. C. y x (Correct answer)
4. D. Cannot be determined

Correct Answer

Option C — y x

Detailed Solution & Explanation

The correct answer is y > x.

Key Points

• Estimate each surd rather than trying to compare the expressions symbolically: $\sqrt{3} \approx 1.732$, $\sqrt{5} \approx 2.236$, so $x \approx 3.968$.
• $\sqrt{8} \approx 2.828$ and $\sqrt{2} \approx 1.414$, so $y \approx 4.243$.
• Since 4.243 exceeds 3.968, y > x — the pair further apart under the radicals wins.

Additional Information

• A cleaner route avoids decimals: square both. $x^2 = 8 + 2\sqrt{15}$ and $y^2 = 10 + 2\sqrt{16} = 10 + 8 = 18$. Since $\sqrt{15} < 4$, $x^2 < 16 < 18$, so $y > x$.
• The underlying principle: for a fixed sum under the radicals, the total $\sqrt{a} + \sqrt{b}$ is larger when a and b are further apart, because the square root is concave.
• Surd values worth memorising: $\sqrt{2} = 1.414$, $\sqrt{3} = 1.732$, $\sqrt{5} = 2.236$, $\sqrt{7} = 2.646$.

प्रश्न (हिन्दी में)

मान लीजिए x = √3 + √5 और y = √8 + √2 । कौन-सा बड़ा है?

1. A. x = y
2. B. x y
3. C. y x
4. D. तय नहीं किया जा सकता