Understanding Square Roots
To find the square root of 10 (√10) accurate to four decimal places, we can use the method of approximation or a calculator. Here, we'll focus on the approximation method for better understanding.

Step-by-Step Approximation
1. **Initial Guess**
- Start by identifying two perfect squares between which 10 lies.
- 3² = 9 and 4² = 16, thus √10 is between 3 and 4.
2. **Refining the Guess**
- Take an initial guess, say 3.1.
- Calculate (3.1)² = 9.61, which is less than 10.
3. **Next Guess**
- Try 3.2: (3.2)² = 10.24, which is greater than 10.
- Therefore, √10 is between 3.1 and 3.2.
4. **Narrowing It Down**
- Try 3.15: (3.15)² = 9.9225, still less than 10.
- Now, try 3.16: (3.16)² = 10.0356, which is greater than 10.
- So, √10 is between 3.15 and 3.16.
5. **Further Refinement**
- Test 3.1625: (3.1625)² = 10.00390625, slightly greater than 10.
- Test 3.162: (3.162)² = 10.005044, also greater than 10.
- Test 3.1615: (3.1615)² = 9.99932225, slightly less than 10.

Final Value
Based on the above steps, we can conclude that:
- The value of √10 is approximately **3.1623**, correct to four decimal places.
Using this method provides a systematic approach to finding square roots through estimation and refinement. This can be particularly useful for competitive exams like UPSC.