Find the square root of :
`0.023104`

Answer

To find the square root of \(0.023104\) using the long division method, we can follow these steps:

### Step 1: Pair the digits
First, we will pair the digits of the number starting from the decimal point. We can write \(0.023104\) as \(0.02 | 31 | 04\).

### Step 2: Set up for long division
We will set up for the long division method. The first pair is \(0\) (from the left of the decimal point).

### Step 3: Find the largest square
The largest square less than or equal to \(0\) is \(0\). So, we write \(0\) above the line.

\[ \begin{array}{r|l} 0 & 0.023104 \\ \hline 0 & 0 \\ \end{array} \]

### Step 4: Subtract and bring down the next pair
Subtract \(0\) from \(0\) to get \(0\), and then bring down the next pair \(02\).

### Step 5: Find the next digit
Now we have \(02\). The next digit is \(0\). We double the number above the line (which is \(0\)), giving us \(0\). We need to find a digit \(x\) such that \(0x \times x \leq 2\). The largest \(x\) is \(1\) because \(1 \times 1 = 1\).

\[ \begin{array}{r|l} 0.1 & 0.023104 \\ \hline 0 & 0 \\ 1 & 1 \\ \end{array} \]

### Step 6: Subtract and bring down the next pair
Subtract \(1\) from \(2\) to get \(1\), and bring down the next pair \(31\), making it \(131\).

### Step 7: Find the next digit
Now we have \(131\). We double the number above the line (which is \(1\)), giving us \(2\). We need to find a digit \(y\) such that \(2y \times y \leq 131\). Testing \(y = 5\):

\[ 25 \times 5 = 125 \quad (\text{which is less than } 131) \]

So we write \(5\) above the line.

\[ \begin{array}{r|l} 0.15 & 0.023104 \\ \hline 0 & 0 \\ 1 & 1 \\ 25 & 125 \\ \end{array} \]

### Step 8: Subtract and bring down the next pair
Subtract \(125\) from \(131\) to get \(6\), and bring down the next pair \(04\), making it \(604\).

### Step 9: Find the next digit
Now we have \(604\). We double the number above the line (which is \(15\)), giving us \(30\). We need to find a digit \(z\) such that \(30z \times z \leq 604\). Testing \(z = 2\):

\[ 302 \times 2 = 604 \quad (\text{which is equal to } 604) \]

So we write \(2\) above the line.

\[ \begin{array}{r|l} 0.152 & 0.023104 \\ \hline 0 & 0 \\ 1 & 1 \\ 25 & 125 \\ 302 & 604 \\ \end{array} \]

### Final Result
After completing the long division, we find that the square root of \(0.023104\) is \(0.152\).