Find the square root of 144.

Answer

To find the square root of 144, we can use the long division method. Here’s a step-by-step solution:

### Step 1: Pair the digits
First, we pair the digits of 144 from right to left. Since 144 has three digits, we can pair them as follows:
- Pair: (1)(44)

### Step 2: Find the largest number whose square is less than or equal to the first pair
We look at the first pair, which is 1. The largest number whose square is less than or equal to 1 is 1 itself (since \(1^2 = 1\)).

### Step 3: Subtract and bring down the next pair
Now, we subtract \(1^2\) from 1:
- \(1 - 1 = 0\)

Next, we bring down the next pair (44), making it 044 (or just 44).

### Step 4: Double the quotient and find the next digit
Now, we double the quotient we found (which is 1) to get 2. We need to find a digit \(x\) such that \(2x \times x\) is less than or equal to 44.

Let’s try \(x = 2\):
- \(2 \times 2 = 4\)
- \(2 \times 2 \times 2 = 8\) (which is less than 44)

Now we try \(x = 3\):
- \(2 \times 3 = 6\)
- \(6 \times 3 = 18\) (which is still less than 44)

Now we try \(x = 4\):
- \(2 \times 4 = 8\)
- \(8 \times 4 = 32\) (which is still less than 44)

Now we try \(x = 5\):
- \(2 \times 5 = 10\)
- \(10 \times 5 = 50\) (which is greater than 44)

So, the largest digit we can use is 4.

### Step 5: Subtract and find the remainder
Now we subtract \(32\) from \(44\):
- \(44 - 32 = 12\)

### Step 6: Bring down the next pair (if any)
Since there are no more pairs to bring down, we can stop here.

### Step 7: Combine the results
The quotient we have obtained is 12. Therefore, the square root of 144 is 12.

### Final Verification
To verify, we can square 12:
- \(12 \times 12 = 144\)

Thus, the square root of 144 is indeed 12.