Find the square root of 3481 by division method.

To find the square root of 3481 using the division method, we can break it down into a series of logical steps. This method is systematic and can be quite effective for larger numbers. Let’s go through it step by step.

Step-by-Step Process

1. Pairing the Digits

First, we need to pair the digits of the number starting from the right. For 3481, we can pair it as follows:

• 34 | 81

This means we have two pairs: 34 and 81.

2. Finding the Largest Square

Next, we look at the leftmost pair (34) and find the largest square number less than or equal to 34. The largest square is 5² = 25.

Now, we write 5 above the line (this is part of our answer) and subtract 25 from 34:

• 34 - 25 = 9

We then bring down the next pair (81) next to 9, giving us 981.

3. Doubling the Quotient

Now, we double the number we have above the line (which is 5) to get 10. This will help us find the next digit of our answer.

4. Finding the Next Digit

We need to find a digit (let’s call it x) such that:

• (10x + x) * x ≤ 981

This simplifies to:

• (10x + x) * x = (11x) * x = 11x²

Now we can test values for x:

• If x = 8: 11 * 8² = 11 * 64 = 704 (which is less than 981)
• If x = 9: 11 * 9² = 11 * 81 = 891 (which is also less than 981)
• If x = 10: 11 * 10² = 11 * 100 = 1100 (which is greater than 981)

The largest value that works is x = 9.

5. Completing the Calculation

Now we place 9 next to 5 above the line, making our current answer 59. We then calculate:

• 11 * 9 = 99
• 981 - 891 = 90

Next, we bring down the next pair of zeros (since we are looking for a decimal), making it 9000.

6. Finding the Next Digit Again

We double our current answer (59) to get 118. Now we need to find a digit (let’s call it y) such that:

• (118y + y) * y ≤ 9000

This simplifies to:

• (119y) * y = 119y²

Testing values for y:

• If y = 7: 119 * 7² = 119 * 49 = 5831 (which is less than 9000)
• If y = 8: 119 * 8² = 119 * 64 = 7616 (which is also less than 9000)
• If y = 9: 119 * 9² = 119 * 81 = 9639 (which is greater than 9000)

The largest value that works is y = 8.

Final Steps

Now we place 8 next to 59, making our answer 58. We calculate:

• 119 * 8 = 952
• 9000 - 952 = 48

Since we have reached a point where we can stop, we can conclude that the square root of 3481 is approximately 59.

Summary

Using the division method, we determined that the square root of 3481 is 59. This method is particularly useful for larger numbers and helps reinforce the understanding of squares and roots through a structured approach.