Find the square root of 12.25 using a long division method.

A. 3.2

B. 3.4

C. 3.5

D. 3.1

To find the square root of 12.25 using the long division method, let’s break it down step by step. This method is quite systematic and can be very effective for finding square roots of numbers that are not perfect squares.

Step-by-Step Long Division Method

1. Set Up the Problem

First, we need to express 12.25 in a way that makes it easier to work with. We can treat it as 1225 by moving the decimal point two places to the right. This means we will find the square root of 1225 and then place the decimal point back in the correct position later.

2. Pair the Digits

Next, we will pair the digits of 1225 from right to left. This gives us the pairs: 12 and 25.

3. Find the Largest Square

Now, we look for the largest square number less than or equal to the first pair (12). The largest square is 3, since \(3^2 = 9\). We write 3 above the line:

• 3 (the quotient)
• Subtract \(9\) from \(12\) to get \(3\).

4. Bring Down the Next Pair

Next, we bring down the next pair (25), making our new number 325.

5. Double the Quotient

Now, we double the quotient we found (3), which gives us 6. We will use this to find the next digit of our answer:

• Let’s denote the next digit as \(x\). We need to find \(x\) such that \(6x \times x\) is less than or equal to 325.

6. Test Possible Values for x

We can try different values for \(x\):

• If \(x = 5\): \(65 \times 5 = 325\) (this works perfectly).
• If \(x = 6\): \(66 \times 6 = 396\) (this is too high).

Since \(x = 5\) works, we write 5 above the line next to 3, making our current quotient 3.5.

7. Final Steps

Now, we subtract \(325\) from \(325\) to get \(0\). Since we have no remainder and have completed the long division, we can conclude that the square root of 1225 is 35.

8. Place the Decimal

Since we initially moved the decimal point two places to the right, we need to move it back two places in our final answer. Thus, the square root of 12.25 is 3.5.

Final Answer

The correct option is C. 3.5.