Find the square root of each of the following correct to two decimal places :
(i) `3(4)/(5)`
(ii) `6(7)/(8)`

Answer

To find the square root of the given numbers correct to two decimal places, we will follow a systematic approach for each part of the question.

### (i) Finding the square root of \( \frac{19}{5} \)

1. **Convert the fraction to a decimal**:
\[ \frac{19}{5} = 3.8 \]

2. **Set up for finding the square root**:
We will find the square root of \( 3.80 \). Since it has one digit before the decimal, we will consider it as \( 3.80 \) (which is \( 3.8000 \)).

3. **Finding the square root using the long division method**:
- Pair the digits: 3 | 80 | 00
- Find the largest number whose square is less than or equal to 3. This is 1 (since \( 1^2 = 1 \)).
- Subtract \( 1^2 \) from 3:
\[ 3 - 1 = 2 \]
- Bring down the next pair (80):
\[ 280 \]
- Double the quotient (which is 1) to get 2, and find a number \( x \) such that \( (20 + x) \times x \) is less than or equal to 280.
- Trying \( x = 6 \):
\[ (20 + 6) \times 6 = 26 \times 6 = 156 \quad (\text{valid}) \]
- Trying \( x = 7 \):
\[ (20 + 7) \times 7 = 27 \times 7 = 189 \quad (\text{valid}) \]
- Trying \( x = 8 \):
\[ (20 + 8) \times 8 = 28 \times 8 = 224 \quad (\text{valid}) \]
- Trying \( x = 9 \):
\[ (20 + 9) \times 9 = 29 \times 9 = 261 \quad (\text{valid}) \]
- Trying \( x = 10 \):
\[ (20 + 10) \times 10 = 30 \times 10 = 300 \quad (\text{not valid}) \]
- So, we take \( x = 9 \):
\[ 280 - 261 = 19 \]
- Bring down the next pair of zeros (00):
\[ 1900 \]
- Double the quotient (which is 19) to get 38, and find \( y \) such that \( (380 + y) \times y \) is less than or equal to 1900.
- Trying \( y = 4 \):
\[ (380 + 4) \times 4 = 384 \times 4 = 1536 \quad (\text{valid}) \]
- Trying \( y = 5 \):
\[ (380 + 5) \times 5 = 385 \times 5 = 1925 \quad (\text{not valid}) \]
- So, we take \( y = 4 \):
\[ 1900 - 1536 = 364 \]

4. **Final result**:
The square root of \( 3.80 \) is approximately \( 1.94 \) (after rounding to two decimal places).

### (ii) Finding the square root of \( \frac{55}{8} \)

1. **Convert the fraction to a decimal**:
\[ \frac{55}{8} = 6.875 \]

2. **Set up for finding the square root**:
We will find the square root of \( 6.875 \) (which is \( 6.8750 \)).

3. **Finding the square root using the long division method**:
- Pair the digits: 6 | 87 | 50
- Find the largest number whose square is less than or equal to 6. This is 2 (since \( 2^2 = 4 \)).
- Subtract \( 2^2 \) from 6:
\[ 6 - 4 = 2 \]
- Bring down the next pair (87):
\[ 287 \]
- Double the quotient (which is 2) to get 4, and find \( x \) such that \( (40 + x) \times x \) is less than or equal to 287.
- Trying \( x = 6 \):
\[ (40 + 6) \times 6 = 46 \times 6 = 276 \quad (\text{valid}) \]
- Trying \( x = 7 \):
\[ (40 + 7) \times 7 = 47 \times 7 = 329 \quad (\text{not valid}) \]
- So, we take \( x = 6 \):
\[ 287 - 276 = 11 \]
- Bring down the next pair of zeros (00):
\[ 1100 \]
- Double the quotient (which is 26) to get 52, and find \( y \) such that \( (520 + y) \times y \) is less than or equal to 1100.
- Trying \( y = 2 \):
\[ (520 + 2) \times 2 = 522 \times 2 = 1044 \quad (\text{valid}) \]
- Trying \( y = 3 \):
\[ (520 + 3) \times 3 = 523 \times 3 = 1569 \quad (\text{not valid}) \]
- So, we take \( y = 2 \):
\[ 1100 - 1044 = 56 \]

4. **Final result**:
The square root of \( 6.875 \) is approximately \( 2.62 \) (after rounding to two decimal places).

### Summary of Results:
- The square root of \( \frac{19}{5} \) is approximately \( 1.94 \).
- The square root of \( \frac{55}{8} \) is approximately \( 2.62 \).