Finding Square Root by Division Method

Finding square root by division method

When the numbers are large, even the method of finding square root by prime factorisation becomes lengthy and difficult. To overcome this problem we use Long Division Method.

For this we need to determine the number of digits in the square root. See the following table:

Number	Square	Property
10	100	which is the smallest -digit perfect square
31	961	which is the greatest -digit perfect square
32	1024	which is the smallest -digit perfect square
99	9801	which is the greatest -digit perfect square

So, what can we say about the number of digits in the square root if a perfect square is a 3-digit or a 4-digit number? We can say that, if a perfect square is a 3-digit or a 4-digit number, then its square root will have digits.

Can you tell the number of digits in the square root of a 5-digit or a 6-digit perfect square?

Instructions

Finding smallest and highest 5 and 6-digit number

The smallest 5-digit perfect square is 10000 which gives 10000 = while the largest 5-digit perfect square is 99856 which gives 99856 =
This tells us that the square root has digits.
Checking for 6-digit perfect square: we have 100489 which gives 100489 = which is the smallest perfect square while the largest 5-digit perfect square is 998001 which gives 998001 =
This tells us that in this case as well, the square root has digits.
Thus, the square root of 5 (or) 6-digit perfect square has a total of 3 digits.

The smallest 3-digit perfect square number is 100 which is the square of 10 and the greatest 3-digit perfect square number is 961 which is the square of 31. The smallest 4-digit square number is 1024 which is the square of 32 and the greatest 4-digit number is 9801 which is the square of 99.

Think Discuss and Write

Can we say that if a perfect square is of n-digits, then its square root will have n2 digits if n is even or n+12 if n is odd?

The relationship between the number of digits in a perfect square and the number of digits in its square root follows a specific pattern, but it does not fit perfectly into the formulas n2 or n+12 for all cases.

If n is even: A perfect square with n digits typically has a square root with n2 digits.

Example:

10000 (a 5-digit perfect square) has 1002. Here, 100 is a -digit number.

1000000 (a 7-digit perfect square) has 10002. Here, 1000 is a -digit number.

If n is odd:

A perfect square with n digits typically has n+12 digits.

The formula you suggested is generally correct but can be considered an approximation. The exact number of digits can vary slightly depending on the specific numbers involved.

The use of the number of digits in square root of a number is useful in the following method: Consider the following steps to find the square root of 529.

Instructions

Finding the square root of a number using long division method

Starting from right place a bar on top of every pair of digits. If a single digit remains on the left, add a bar to that as well. For eg: 5 29
From the left, find the square which is less than or equal to the digit. Here we have, (22 < 5 < 32). Here, 2 becomes the first digit of the quotient and we get a remainder of when subtracting 22 from 5.
Bring down the number under the next bar (i.e. 29) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 4_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
As × 3 = 129, we can choose the digit to be filled in the blank as .
Since, the remainder is and there are no more digits left in the given number, we get: 529 =
We have found the square root of 529.

Now consider 4096

Instructions

Finding the square root of a number using long division method

Place a bar over every pair of digits starting from the one’s digit. (40 96)
From the left, find the square which is less than or equal to the digit. Here we have, (62 < 40 < 72). Here, 6 becomes the first digit of the quotient and we get a remainder of when subtracting 62 from 40.
Bring down the number under the next bar (i.e. 96) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 12_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
We get that: 124 × = 496
Since, the remainder is and there are no more digits left in the given number, we get: 4096 = .
We have found the square root of 4096.

Estimating the number

We use bars to find the number of digits in the square root of a perfect square number.

= 23 and = 64

In both the numbers 529 and 4096 there are two bars and the number of digits in their square root is .

Can you tell the number of digits in the square root of 14400 ? By placing bars on top of , we get bars.

Thus, the square root will be having 3 digits.

Without calculating square roots, find the number of digits in the square root of the following numbers.

Instructions

If the number of digits n is , the square root will have n2 digits. If the number of digits n is , the square root will have n+12 digits.

(i) 25600

Number of digits in 25600 = i.e. . Thus, the square root will have 5+12 = digits.

(ii) 100000000

Number of digits in 100000000 = i.e. . Thus, the square root will have 9+12 = digits.

(iii) 36864

Number of digits in 36864 = i.e. . Thus, the square root will have 5+12 = digits.

Example 9: Find the square root of:

Instructions

(i) 729

Finding the square root using long division method

Place a bar over every pair of digits starting from the one’s digit. (7 29)
From the left, find the square which is less than or equal to the digit. Here we have, (22 < 7 < 32). Here, 2 becomes the first digit of the quotient and we get a remainder of when subtracting 22 from 7.
Bring down the number under the next bar (i.e. 29) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 4_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
We get that: 47 × = 329.
Since, the remainder is and there are no more digits left in the given number, we get: 729 = .
We have found the square root of 729.

(ii) 1296

Instructions

Finding the square root of a number using long division method

Place a bar over every pair of digits starting from the one’s digit. (12 96)
From the left, find the square which is less than or equal to the digit. Here we have, (32 < 12 < 42). Here, 3 becomes the first digit of the quotient and we get a remainder of when subtracting 32 from 12.
Bring down the number under the next bar (i.e. 96) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 6_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
We get that: 66 × = 396
Since, the remainder is and there are no more digits left in the given number, we get: 1296 = .
We have found the square root of 1296.

Example 10: Find the least number that must be subtracted from 5607 so as to get a perfect square. Also find the square root of the perfect square.

Instructions

Finding sqrt(5607)

Place a bar over every pair of digits starting from the one’s digit. (56 07)
From the left, find the square which is less than or equal to the digit. Here we have, (72 < 56 < 82). Here, 7 becomes the first digit of the quotient and we get a remainder of when subtracting 72 from 56.
Bring down the number under the next bar (i.e.076) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 14_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
We get that: 144 × = 576
Since, the remainder is and quotient is . Thus 742 5607.
Thus, the least number that must be subtracted from 5607 to get a perfect square: .

Example 11: Find the greatest 4-digit number which is a perfect square.

Instructions

Finding the largest 4 digit perfect square

Let's take the greatest 4-digit number i.e.
Place a bar over every pair of digits starting from the one’s digit. (99 99)
From the left, find the square which is less than or equal to the digit. Here we have, (92 < 99 < 102). Here, 9 becomes the first digit of the quotient and we get a remainder of when subtracting 92 from 99.
Bring down the number under the next bar (i.e. 99) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 18_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
We get that: 189 × = 1701
Since, the remainder is and there are no more digits left in the given number, we can say that we need to subtract from in order to get the largest four-digit perfect square. Thus, the answer is
We also find that 9801 = .

Example 12: Find the least number that must be added to 1300 so as to get a perfect square. Also find the square root of the perfect square.

Instructions

Finding sqrt(1300)

Place a bar over every pair of digits starting from the one’s digit. (13 00)
From the left, find the square which is less than or equal to the digit. Here we have, (32 < 13 < 42). Here, 3 becomes the first digit of the quotient and we get a remainder of when subtracting 32 from 13.
Bring down the number under the next bar (i.e. 00) to the right of the remainder. We now get the new dividend i.e.
In the divisor put the first digit as double the quotient and place a blank its right. i.e. 6_
Put the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
We get that: 66 × = 396
Since, the remainder is and there are no more digits left in the given number.
Thus, we have as our quotient which tells us that 362 < 1300. We know that 372 = .
Thus, the least number that must be added to 1300 so as to get a perfect square: 1369 - 1300 = .

Chapter 6: Square Roots and Cube Roots

Glossary

Finding Square Root by Division Method