Find the square root of `7.832` corret to :
(i) `2` decimal places
(ii) `2` significant digits.

Answer

To find the square root of \( 7.832 \) correct to two decimal places and two significant digits, we can follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**:
- Start by writing \( 7.832 \) as \( 7.832000 \) to make it easier to pair the digits.
- Pair the digits starting from the decimal point: \( (7)(83)(20)(00) \).

2. **Find the Largest Square**:
- The first pair is \( 7 \). The largest square less than or equal to \( 7 \) is \( 2^2 = 4 \).
- Write \( 2 \) above the line and subtract \( 4 \) from \( 7 \):
\[ 7 - 4 = 3 \]

3. **Bring Down the Next Pair**:
- Bring down the next pair \( 83 \) next to \( 3 \), making it \( 383 \).

4. **Double the Quotient**:
- Double the current quotient \( 2 \) to get \( 4 \). Now, we need to find a digit \( x \) such that \( (40 + x)x \) is less than or equal to \( 383 \).
- Testing \( x = 7 \):
\[ (40 + 7) \times 7 = 47 \times 7 = 329 \]
- This is less than \( 383 \).

5. **Subtract and Bring Down the Next Pair**:
- Subtract \( 329 \) from \( 383 \):
\[ 383 - 329 = 54 \]
- Bring down the next pair \( 20 \), making it \( 5420 \).

6. **Repeat the Process**:
- Double the current quotient \( 27 \) to get \( 54 \). Now we need to find \( y \) such that \( (540 + y)y \) is less than or equal to \( 5420 \).
- Testing \( y = 9 \):
\[ (540 + 9) \times 9 = 549 \times 9 = 4941 \]
- This is less than \( 5420 \).

7. **Subtract Again**:
- Subtract \( 4941 \) from \( 5420 \):
\[ 5420 - 4941 = 479 \]
- Bring down the next pair \( 00 \), making it \( 47900 \).

8. **Continue the Process**:
- Double the current quotient \( 279 \) to get \( 558 \). Now find \( z \) such that \( (5580 + z)z \) is less than or equal to \( 47900 \).
- Testing \( z = 8 \):
\[ (5580 + 8) \times 8 = 5588 \times 8 = 44704 \]
- This is less than \( 47900 \).

9. **Final Subtraction**:
- Subtract \( 44704 \) from \( 47900 \):
\[ 47900 - 44704 = 3196 \]

10. **Result**:
- After performing these steps, we find that the square root of \( 7.832 \) is approximately \( 2.798 \).
- Since the last digit \( 8 \) is more than \( 5 \), we round up the previous digit:
\[ \text{Square root of } 7.832 \text{ correct to two decimal places is } 2.80. \]

### Significant Digits:
- The significant digits are the non-zero digits. Here, the first two significant digits are \( 2 \) and \( 8 \).
- Therefore, the square root of \( 7.832 \) correct to two significant digits is \( 2.8 \).

### Final Answers:
(i) Square root of \( 7.832 \) correct to two decimal places: **2.80**
(ii) Square root of \( 7.832 \) correct to two significant digits: **2.8**