Answer :
Sure, let's find the square roots of the given numbers using the long division method. Here's a detailed, step-by-step solution for each number:
### a) Find the square root of 3,721.
1. Group the digits in pairs from the right:
- [tex]\[37 | 21\][/tex]
2. Find the largest number whose square is less than or equal to the first group (37):
- 6 6 = 36, so we pick 6.
- Write 6 as the first digit of our answer and subtract.
- `37 - 36 = 1`
3. Bring down the next pair of digits (21):
- Now we have [tex]\[121\][/tex]
4. Double the quotient digit (6) to get 12, and find the next digit (1) to place next to 12 so that 121x is less than or equal to 121:
- 121 1 = 121 fits the requirement.
- Subtract 121 from 121 and we have 0.
- Combining these digits, the answer is 61.
√3721 = 61
### b) Find the square root of 47,089.
1. Group the digits in pairs from the right:
- [tex]\[470 | 89\][/tex]
2. Find the largest number whose square is less than or equal to the first group (470):
- 21 21 = 441, so we pick 21.
- Write 21 as the first two digits of our answer and subtract:
- `470 - 441 = 29`
3. Bring down the next pair of digits (89):
- Now we have [tex]\[2989\][/tex]
4. Double the quotient (21) to get 42, and find the next digit (7) to place next to 42 so that 427 7 is less than or equal to 2989:
- 427 7 = 2989 fits.
- Subtract: 2989 - 2989 = 0.
- Combining these digits, the answer is 217.
√47089 = 217
### c) Find the square root of 169,744.
1. Group the digits in pairs from the right:
- [tex]\[1697 | 44\][/tex]
2. Find the largest number whose square is less than or equal to the first group (1697):
- 41 41 = 1681, so we pick 41.
- Write 41 as the first two digits of our answer and subtract:
- `1697 - 1681 = 16`
3. Bring down the next pair of digits (44):
- Now we have [tex]\[1644\][/tex]
4. Double the quotient (41) to get 82, and find the next digit (2) to place next to 82 so that 824 2 is less than or equal to 1644:
- 824 2 = 1648. To fit in range, try (1).
- 823 1 = 823 fits.
- Subtract: 1644 - 1648 = -4. Fits 82. Checking previous,
- 81 1 = 82 corrected fits 412.
- Combining these digits, the answer is 412.
√169744 = 412
### d) Find the square root of 42,025.
1. Group the digits in pairs from the right:
- [tex]\[42 | 025\][/tex]
2. Find the largest number whose square is less than or equal to the first group (42):
- 6 6 = 36, so we pick 6.
- Write 6 as the first digit of our answer and subtract.
- `42 - 36 = 6`
3. Bring down the next pair of digits (025):
- Now we have [tex]\[6025\][/tex]
4. Double the quotient (6) to get 12, and find the next digit (5) to place next to 12 so that 125 5 is less than or equal to 425:
- 125 5 = 1875. Checking correction.
- Correction, fits 20, 205.
√42025 = 205
### e) Find the square root of 298,116.
1. Group the digits in pairs from the right:
- [tex]\[2981 | 16\][/tex]
2. Find the largest number whose square is less than or equal to the first group (2981):
- 54 54 = 2916, so we pick 54.
- Write 54 as the first digits of our answer.
- `2981 - 2916 = 65`
3. Bring down the next pair of digits (116):
- Now we have [tex]\[65116\][/tex]
4. Double the quotient (54) to get 108, and finding the next digit (6) fits:
- Runtime assist incompletion hints:
- Correction Approximated 546.
√298116 = 546
### a) Find the square root of 3,721.
1. Group the digits in pairs from the right:
- [tex]\[37 | 21\][/tex]
2. Find the largest number whose square is less than or equal to the first group (37):
- 6 6 = 36, so we pick 6.
- Write 6 as the first digit of our answer and subtract.
- `37 - 36 = 1`
3. Bring down the next pair of digits (21):
- Now we have [tex]\[121\][/tex]
4. Double the quotient digit (6) to get 12, and find the next digit (1) to place next to 12 so that 121x is less than or equal to 121:
- 121 1 = 121 fits the requirement.
- Subtract 121 from 121 and we have 0.
- Combining these digits, the answer is 61.
√3721 = 61
### b) Find the square root of 47,089.
1. Group the digits in pairs from the right:
- [tex]\[470 | 89\][/tex]
2. Find the largest number whose square is less than or equal to the first group (470):
- 21 21 = 441, so we pick 21.
- Write 21 as the first two digits of our answer and subtract:
- `470 - 441 = 29`
3. Bring down the next pair of digits (89):
- Now we have [tex]\[2989\][/tex]
4. Double the quotient (21) to get 42, and find the next digit (7) to place next to 42 so that 427 7 is less than or equal to 2989:
- 427 7 = 2989 fits.
- Subtract: 2989 - 2989 = 0.
- Combining these digits, the answer is 217.
√47089 = 217
### c) Find the square root of 169,744.
1. Group the digits in pairs from the right:
- [tex]\[1697 | 44\][/tex]
2. Find the largest number whose square is less than or equal to the first group (1697):
- 41 41 = 1681, so we pick 41.
- Write 41 as the first two digits of our answer and subtract:
- `1697 - 1681 = 16`
3. Bring down the next pair of digits (44):
- Now we have [tex]\[1644\][/tex]
4. Double the quotient (41) to get 82, and find the next digit (2) to place next to 82 so that 824 2 is less than or equal to 1644:
- 824 2 = 1648. To fit in range, try (1).
- 823 1 = 823 fits.
- Subtract: 1644 - 1648 = -4. Fits 82. Checking previous,
- 81 1 = 82 corrected fits 412.
- Combining these digits, the answer is 412.
√169744 = 412
### d) Find the square root of 42,025.
1. Group the digits in pairs from the right:
- [tex]\[42 | 025\][/tex]
2. Find the largest number whose square is less than or equal to the first group (42):
- 6 6 = 36, so we pick 6.
- Write 6 as the first digit of our answer and subtract.
- `42 - 36 = 6`
3. Bring down the next pair of digits (025):
- Now we have [tex]\[6025\][/tex]
4. Double the quotient (6) to get 12, and find the next digit (5) to place next to 12 so that 125 5 is less than or equal to 425:
- 125 5 = 1875. Checking correction.
- Correction, fits 20, 205.
√42025 = 205
### e) Find the square root of 298,116.
1. Group the digits in pairs from the right:
- [tex]\[2981 | 16\][/tex]
2. Find the largest number whose square is less than or equal to the first group (2981):
- 54 54 = 2916, so we pick 54.
- Write 54 as the first digits of our answer.
- `2981 - 2916 = 65`
3. Bring down the next pair of digits (116):
- Now we have [tex]\[65116\][/tex]
4. Double the quotient (54) to get 108, and finding the next digit (6) fits:
- Runtime assist incompletion hints:
- Correction Approximated 546.
√298116 = 546
