To find the square root of a perfect square by using the long division method is easy when the numbers are very large since, the method of finding their square roots by factorization becomes lengthy and difficult.

**Step I:** Group the digits in pairs, starting with the digit in the units place. Each pair and the remaining digit (if any) is called a period.

**Step II:** Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor and also as the quotient.

**Step III:** Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.

**Step IV:** Now, the new divisor is obtained by taking two times the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend. **Step V:** Repeat steps (2), (3) and (4) till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.

*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.

*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.

*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.

*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

● **Square Root**

**Square Root of a Perfect Square by using the Prime Factorization Method**

**Square Root of a Perfect Square by Using the Long Division Method**

**Square Root of Numbers in the Decimal Form**

**Square Root of Number in the Fraction Form**

**Square Root of Numbers that are Not Perfect Squares**

**Practice Test on Square and Square Roots**

● **Square Root- Worksheets**

**Worksheet on Square Root using Prime Factorization Method**

**Worksheet on Square Root using Long Division Method**

**Worksheet on Square Root of Numbers in Decimal and Fraction Form**

**8th Grade Math Practice** **From Square Root of a Perfect Square by Using the Long Division Method to HOME PAGE**

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