find square root by long division method

find square root by long division method 6561. 390625. 108241. 363609

The Correct Answer and Explanation is:

Square Root by Long Division Method

The long division method involves finding the square root of a number by pairing the digits in groups of two, starting from the decimal point (or from the right in case of an integer). You then perform a series of division and subtraction steps to estimate the square root.

Let’s break down the square roots of the numbers you provided:

1. Square root of 6561

1. Start by pairing the digits: 65 | 61.
2. Find the largest number whose square is less than or equal to the first pair (65). The largest such number is 8 because 8² = 64.
3. Subtract: 65 – 64 = 1. Bring down the next pair (61) to make 161.
4. Double the number 8 (the divisor), and add a digit (denote it as d), which when added to 16d, will be as close as possible to 161.
   • The closest value is 81 because 81 × 1 = 81. Subtract: 161 – 81 = 80.
5. Repeat, but you will soon realize that 81 is the exact square root. Hence, the square root of 6561 is 81.

2. Square root of 390625

1. Pair the digits: 39 | 06 | 25.
2. Find the largest number whose square is less than or equal to 39. The largest number is 6 because 6² = 36.
3. Subtract: 39 – 36 = 3. Bring down the next pair (06) to get 306.
4. Double the number 6, giving 12. Now find a digit d such that (120 + d) × d is as close to 306.
   • The digit is 2 because 122 × 2 = 244. Subtract: 306 – 244 = 62.
5. Bring down the next pair (25) to make 625.
6. Double 12 (the divisor), and find a digit d such that (240 + d) × d is as close as possible to 625.
   • The digit is 5 because 245 × 5 = 1225.
7. The square root of 390625 is 625.

3. Square root of 108241

1. Pair the digits: 10 | 82 | 41.
2. Find the largest number whose square is less than or equal to 10. The largest number is 3 because 3² = 9.
3. Subtract: 10 – 9 = 1. Bring down the next pair (82) to make 182.
4. Double 3, giving 6. Find a digit d such that (60 + d) × d is as close as possible to 182.
   • The digit is 2 because 62 × 2 = 124. Subtract: 182 – 124 = 58.
5. Bring down the next pair (41) to make 5841.
6. Double 62 and find a digit d. The correct digit is 1. The square root is 329.

4. Square root of 363609

1. Pair the digits: 36 | 36 | 09.
2. Find the largest number whose square is less than or equal to 36. The largest number is 6 because 6² = 36.
3. Subtract: 36 – 36 = 0. Bring down the next pair (36) to make 36.
4. Double 6, giving 12. Find a digit d such that (120 + d) × d is as close as possible to 36.
   • The digit is 3 because 123 × 3 = 369. Subtract: 36 – 369 = 63.
5. The square root of 363609 is 603.

Summary

• Square root of 6561 is 81.
• Square root of 390625 is 625.
• Square root of 108241 is 329.
• Square root of 363609 is 603.

The long division method is a great way to find square roots manually, especially when the numbers are perfect squares. It ensures accuracy by progressively narrowing down the estimate, and it works for both whole numbers and decimals.