How to Find Square Root by Division Method

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How to Find Square root by division method: Any number can be shown as an output of prime numbers. This technique of illustration of a number in terms of the product of prime numbers is termed the prime factorization method. It is the simple technique known for the standard calculation of the square root of a number. But this technique becomes boring and tired when the number involved is large. Therefore, to find the square root of large numbers we use the long division method.

Also, to find the square roots of imperfect squares such as 2,3,5,6,8,etc., we can use long division method escape the use of calculators.

The square root of a number by long division method

  • Step I: Group the digits two by two, beginning with the digit in the unit’s place. Each pair and the extra digit (assuming any) is known as a period.
  • Step II: Think of the biggest number whose square is equivalent to or only not exactly the primary period. Accept this number as the divisor and furthermore as the remainder.
  • Step III: Subtract the result of the divisor and the remainder from the primary time frame and cut down the following period to one side of the rest. This turns into a new profit.
  • Step IV: Now, the new divisor is acquired by taking multiple times the remainder and attaching with it a reasonable digit which is likewise taken as the following digit of the remainder, picked so that the result of the new divisor and this digit is equivalent to or only not exactly the new profit.
  • Step V: Repeat steps (2), (3) and (4) till every one of the time frames have been taken up. Presently, the remainder so got is the necessary square foundation of the given number.

How to Find Square Root by Division Method

Let us understand long division method with the help of an example.

  1. Taking 484 as the number whose square root is to be assessed. Spot a bar over the pair of numbers beginning from the unit spot or Right-hand side of the number. In the event that, in the event that we have the complete number of digits as an odd number, the furthest left digit will likewise have a bar, 4¯84¯.
  2. Take the biggest number as the divisor whose square is not exactly or equivalent to the number on the limit left of the number. The digit on the limit left is the profit. Gap and compose the remainder. Here the remainder is 2 and the rest of 0.
  1. Next, we at that point cut down the number, which is under the bar, to the correct side of the rest. Here, for this situation, we cut down 84. Presently, 84 is our new profit.
  • Now double the value of the quotient and enter it with blank space on the right side.
    long division method
  1. Next, we need to choose the biggest digit for the unit spot of the divisor (4_) to such an extent that the new number, when increased by the new digit at the unit’s place, is equivalent to or not exactly the profit (84).

In this case,  42 × 2 = 84. So the new digit is 2.

  1. The remainder is 0, and we have no number left for division, therefore, 484−−−√ = 22.

How to Find Square Root by Division Method

Examples of the square root of a perfect square by using the long division method

1. Find the square root of 784 by the long-division method.

Solution:


Marking periods and using the long-division method,

Therefore, √784 = 28

2. Evaluate √5329 using long-division method.

Solution:


Marking periods and using the long-division method,

Therefore, √5329 =73

3. Evaluate: √16384.

Solution:


Marking periods and using the long-division method,

Therefore, √16384 = 128.

How to Find Square Root by Division Method

4. Evaluate: √10609.

Solution:


Marking periods and using the long-division method,

Therefore, √10609 = 103

5. Evaluate: √66049.

Solution:


Marking periods and using the long-division method,

Therefore, √66049 = 257

6. Find the cost of erecting a fence around a square field whose area is 9 hectares if fencing costs $ 3.50 per metre.

Solution:


Area of the square field = (9 × 1 0000) m² = 90000 m²

Length of each side of the field = √90000 m = 300 m.

Perimeter of the field = (4 × 300) m = 1200 m.

Cost of fencing = $(1200 × ⁷/₂) = $4200.

How to Find Square Root by Division Method

7. Find the least number that must be added to 6412 to make it a perfect square.

Solution:


We try to find out the square root of 6412.

We observe here that (80)² < 6412 < (81)²

The required number to be added = (81)² – 6412

= 6561 – 6412

= 149

Therefore, 149 must be added to 6412 to make it a perfect square.

8. What least number must be subtracted from 7250 to get a perfect square? Also, find the square root of this perfect square.

Solution:


Let us try to find the square root of 7250.

This shows that (85)² is less than 7250 by 25. 


So, the least number to be subtracted from 7250 is 25.

Required perfect square number = (7250 – 25) = 7225

And, √7225 = 85.

9. Find the greatest number of four digits which is a perfect square.

Solution


Greatest number of four digits = 9999.

Let us try to find the square root of 9999.

This shows that (99)² is less than 9999 by 198. 
So, the least number to be subtracted is 198.

Hence, the required number is (9999 – 198) = 9801.

How to Find Square Root by Division Method

10. What least number must be added to 5607 to make the sum a perfect square? Find this perfect square and its square root.

Solution:


We try to find out the square root of 5607.

We observe here that (74)² < 5607 < (75)²

The required number to be added = (75)² – 5607

= (5625 – 5607) = 18

11. Find the least number of six digits which is a perfect square. Find the square root of this number.

Solution:


The least number of six digits = 100000, which is not a perfect square.

Now, we must find the least number which when added to 1 00000 gives a perfect square. This perfect square is the required number.

Now, we find out the square root of 100000.

Clearly, (316)² < 1 00000 < (317)²


Therefore, the least number to be added = (317)² – 100000 = 489.

Hence, the required number = (100000 + 489) = 100489.

Also, √100489 = 317.

12. Find the least number that must be subtracted from 1525 to make it a perfect square.

Solution:


Let us take the square root of 1525

We observe that, 39² < 1525
Therefore, to get a perfect square, 4 must be subtracted from 1525.
Therefore the required perfect square = 1525 – 4 = 1521

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