Which one among the following is the smallest?

Options

A. a)`sqrt401 - sqrt399`
B. b)`sqrt101 - sqrt99`
C. c)`sqrt301 - sqrt299`
D. d)`sqrt201 - sqrt199`

Answer

To determine which among the given options is the smallest, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Options**: Let's say the options are √401, √399, √101, √99, √301, √299, √201, and √199.

2. **Group the Terms**: Notice that the terms can be grouped based on their proximity:
- (√401, √399)
- (√101, √99)
- (√301, √299)
- (√201, √199)

3. **Calculate Differences**: For each pair, calculate the difference between the square roots:
- √401 - √399
- √101 - √99
- √301 - √299
- √201 - √199

4. **Analyze Differences**: The difference between two square roots can be approximated using the formula:
\[ \sqrt{a} - \sqrt{b} \approx \frac{a - b}{\sqrt{a} + \sqrt{b}} \]
This means that the smaller the numbers (a and b), the larger the difference will be.

5. **Apply the Formula**:
- For √401 and √399:
\[ \sqrt{401} - \sqrt{399} \approx \frac{2}{\sqrt{401} + \sqrt{399}} \]
- For √101 and √99:
\[ \sqrt{101} - \sqrt{99} \approx \frac{2}{\sqrt{101} + \sqrt{99}} \]
- For √301 and √299:
\[ \sqrt{301} - \sqrt{299} \approx \frac{2}{\sqrt{301} + \sqrt{299}} \]
- For √201 and √199:
\[ \sqrt{201} - \sqrt{199} \approx \frac{2}{\sqrt{201} + \sqrt{199}} \]

6. **Compare the Results**: Since the differences are inversely proportional to the sum of the square roots, the pairs with smaller numbers will yield larger differences. Thus, the smallest value will be from the pair with the smallest numbers.

7. **Conclusion**: After analyzing the differences, we find that √101 - √99 yields the largest difference, indicating that √101 is the smallest among the options provided.

### Final Answer:
The smallest among the given options is **√99**.