Find the square root of :
`27.3529`

Answer

To find the square root of the number \( 27.3529 \), we will use the long division method for square roots. Here is a step-by-step solution:

### Step 1: Pair the digits
Start by pairing the digits from the decimal point. For \( 27.3529 \), we pair as follows:
- Before the decimal: \( 27 \)
- After the decimal: \( 35 \) and \( 29 \)

### Step 2: Find the largest square
Find the largest square less than or equal to \( 27 \):
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \) (too large)

The largest square is \( 5^2 = 25 \).

### Step 3: Subtract and bring down
Subtract \( 25 \) from \( 27 \):
\[ 27 - 25 = 2 \]
Now, bring down the next pair \( 35 \) to make it \( 235 \).

### Step 4: Double the quotient
Double the current quotient (which is \( 5 \)):
\[ 5 \times 2 = 10 \]
Now we need to find a digit \( x \) such that \( 10x \times x \) is less than or equal to \( 235 \).

### Step 5: Find the next digit
Testing \( x = 2 \):
\[ 102 \times 2 = 204 \quad (which is less than 235) \]
Testing \( x = 3 \):
\[ 103 \times 3 = 309 \quad (which is greater than 235) \]
So, we take \( x = 2 \).

### Step 6: Subtract again
Subtract \( 204 \) from \( 235 \):
\[ 235 - 204 = 31 \]
Now, bring down the next pair \( 29 \) to make it \( 3129 \).

### Step 7: Double the new quotient
Now, double the current quotient (which is \( 52 \)):
\[ 52 \times 2 = 104 \]
Now we need to find a digit \( y \) such that \( 104y \times y \) is less than or equal to \( 3129 \).

### Step 8: Find the next digit
Testing \( y = 3 \):
\[ 1043 \times 3 = 3129 \quad (which is equal to 3129) \]
So, we take \( y = 3 \).

### Step 9: Final subtraction
Subtract \( 3129 \) from \( 3129 \):
\[ 3129 - 3129 = 0 \]
Now we have completed the calculation.

### Final Result
The square root of \( 27.3529 \) is \( 5.23 \).