The square root of 102 up to three places of decimal is…………

Options

A. 10.098
B. 10.099
C. 10.097
D. 10.096

Answer

To find the square root of 102 up to three decimal places, we can use the long division method for square roots. Here’s a step-by-step solution:

### Step 1: Pair the digits
We start by pairing the digits of the number from the decimal point. For 102, we can write it as 1 | 02. This means we have one pair: (1) and (02).

### Step 2: Find the largest square
Next, we find the largest square number less than or equal to the first pair (1). The largest square is \(1^2 = 1\). We write 1 above the square root symbol.

### Step 3: Subtract and bring down the next pair
Subtract \(1\) from \(1\) to get \(0\). Now, bring down the next pair (02) to get \(02\).

### Step 4: Double the current result
Double the number above the square root (which is 1) to get \(2\). This will be the starting point for the next digit.

### Step 5: Find the next digit
Now, we need to find a digit \(x\) such that \(2x \cdot x \leq 2\). Testing \(x = 0\) gives \(0\) and testing \(x = 1\) gives \(2\). So, we can take \(x = 1\).

### Step 6: Write down the next digit
We write \(1\) next to the \(1\) above the square root. Now we have \(1.0\).

### Step 7: Subtract and bring down the next pair
Now, we subtract \(2\) from \(2\) to get \(0\) and bring down the next pair (00) to get \(00\).

### Step 8: Repeat the process
Double the current result (which is now 11) to get \(22\). We now need to find a digit \(y\) such that \(22y \cdot y \leq 0\). The only digit that works here is \(0\).

### Step 9: Write down the next digit
We write \(0\) next to \(1.1\) to get \(1.10\).

### Step 10: Subtract and bring down the next pair
Now we have \(0\) and bring down the next pair (00) to get \(00\).

### Step 11: Continue the process
Double the current result (which is now 110) to get \(220\). We need to find a digit \(z\) such that \(220z \cdot z \leq 0\). The only digit that works here is \(0\).

### Step 12: Write down the next digit
We write \(0\) next to \(1.10\) to get \(1.100\).

### Step 13: Finalize the answer
Now we have reached three decimal places. The square root of 102 up to three decimal places is approximately \(10.099\).

### Final Answer
The square root of 102 up to three decimal places is **10.099**.

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