To calculate the square root of 10 manually, we can use a method called long division or approximation. Here’s a step-by-step approach to get a value for $\sqrt{10}$.

Step-by-Step Process:

1. Estimate the Square Root: We know that:

   • $\sqrt{9} = 3$
   • $\sqrt{16} = 4$

   Since 10 is between 9 and 16, we know $\sqrt{10}$ is between 3 and 4. Let's narrow it down further.

2. Divide 10 by an Approximate Guess: We start by guessing a value, say 3.2, and divide 10 by it: $\frac{10}{3.2} = 3.125$

3. Average the Guess and the Quotient: Now, take the average of 3.2 and 3.125: $\frac{3.2 + 3.125}{2} = 3.1625$

4. Refine the Estimate: Now, repeat the process by dividing 10 by the new guess (3.1625): $\frac{10}{3.1625} \approx 3.1622$

5. Repeat the Process: You can keep repeating this process to get even closer to the square root. After one or two more iterations, you'll arrive at an approximation: $\sqrt{10} \approx 3.162277660168379$

This is a very close estimate. The more iterations you do, the more accurate the result becomes.

Final Answer:

$\sqrt{10} \approx 3.162$

Would you like more details on any of these steps or any other method of calculation?

Related Questions:

1. How do we calculate square roots using a calculator?
2. What is the significance of the square root in real-world problems?
3. Can we calculate square roots of negative numbers? How?
4. What other methods exist for approximating square roots?
5. How is the square root operation used in trigonometry?

Tip:

Using a calculator or a computer program can give you an exact result, but manual approximation methods help in understanding the underlying math.