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The problem asks to:

1. Find the square root of 29.16 using the long division method.
2. Then compute the expression:  
\[
\sqrt{29.16} + \sqrt{0.2916} + \sqrt{2916}
\]

### Step 1: Square root of 29.16 using the long division method
I will guide you through the long division method:

- **Step 1:** Pair the digits starting from the decimal point (29.16 -> 29 | 16).
- **Step 2:** Start with 29. The closest perfect square is 25, whose square root is 5.
- **Step 3:** Subtract 25 from 29, and you get 4. Now bring down the next pair (16) to get 416.
- **Step 4:** Double 5 to get 10, and determine the next digit by finding what makes \(10x \times x\) close to 416. The answer is 4 (since \(104 \times 4 = 416\)).
  
Thus, \(\sqrt{29.16} = 5.4\).

### Step 2: Square root of 0.2916
Since \(0.2916 = (0.54)^2\), the square root of 0.2916 is 0.54.

### Step 3: Square root of 2916
Since \(2916 = 54^2\), the square root of 2916 is 54.

### Step 4: Compute the full expression
Now compute the full expression:
\[
\sqrt{29.16} + \sqrt{0.2916} + \sqrt{2916} = 5.4 + 0.54 + 54 = 59.94
\]

So, the value of the expression is **59.94**.

Would you like a detailed explanation of the long division method? Here are some related questions:

1. What are the steps of the long division method in detail?
2. Can you explain why \( \sqrt{0.2916} = 0.54 \)?
3. How do we estimate square roots for non-perfect squares?
4. Is there an alternative method to find square roots manually?
5. How can you check if your square root calculation is accurate?

**Tip:** Always verify your square root results by squaring them to check if they match the original number!

Find the square root of 29.16 using the long division method and hence find the value of the expression √29.16 + √0.2916 + √2916.

Learn how to find the square root of 29.16 using the long division method and compute the value of the expression √29.16 + √0.2916 + √2916. This step-by-step guide covers essential mathematical concepts and techniques, suitable for students in Grades 8-10.

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