To approximate the square root of 29, we can estimate it by considering the square roots of nearby perfect squares:

• The square root of 25 is 5, and the square root of 36 is 6.
• Since 29 is between 25 and 36, we know that $\sqrt{29}$ is between 5 and 6.

We can refine our estimate further:

Step 1: Try a guess, say $x = 5.4$, and calculate $x^2$:

$5.4^2 = 29.16$

Since 29.16 is a bit larger than 29, try a slightly smaller guess, say $x = 5.3$:

$5.3^2 = 28.09$

Now, since 29 is between 28.09 and 29.16, $\sqrt{29}$ is closer to 5.4.

Step 2: Further refining:

Let's average 5.3 and 5.4, since we know $\sqrt{29}$ is between them.

$\text{Average} = \frac{5.3 + 5.4}{2} = 5.35$

Now check:

$5.35^2 = 28.6225$

Since 28.6225 is still slightly less than 29, we might want to adjust further.

Thus, the square root of 29 is approximately:

$\sqrt{29} \approx 5.385$

This is a reasonable approximation to 3 decimal places.

Would you like more details, or do you have further questions on this? Here's a list of related questions you could explore:

1. How do we calculate square roots using the Newton-Raphson method?
2. What is the difference between exact and approximate square roots?
3. How can I find square roots by hand for other numbers?
4. Can square roots be estimated for negative numbers?
5. Why do we approximate square roots instead of calculating the exact value?

Tip: If you're approximating square roots, it's often helpful to use nearby perfect squares to narrow down the range and refine the estimate step by step.