# Square Roots

16 is a perfect square,

Â

**Example :Â **4^{2}Â = 16.

Here 4 is known as the square root of 16.

Similarly 5 is the square root of 25, 13 is the square root of 169, 25 is the square root of 625.

The square root of a number n is that number which, when multiplied by itself gives n. The square root of n is denoted as The symbol '' means `square root of'. Thus Â Â Â

Square Roots of Perfect Squares

**1. Prime Factorisation Method**

Â

Find the square root of 17424.

Â

Find the least number by which 52 should be multiplied to make it a perfect square.Â

By prime factorisation we getÂ

52 = 2 x 2 x 13

Â

To make this a perfect square, it should be possible to make pairs of prime factors.

We have to multiply 52 with 13 to make it a perfect square.

Thus 52 x 13 =Â __2 x 2__Â xÂ __13 x 13__Â is a perfect square.

Finding the square root of the perfect square ending in 25.

Â |

Finding the square root of a perfect square ending in 25 is the reverse process of the above, as shown in the following examples:

Â

Find the square root of the perfect square 7225.

(i) As the perfect square ends in 25, its square root must end in 5. Write 5 in the units place.

Â

(ii) Find a number, which when multiplied by its successor, gives 72 : 8 x 9 = 72

Â

(iii) 8 must be present in the tens place; hence

Long Division Method

Â

The prime factorisation method becomes very long for large numbers. In such cases we use the long division method, which consists of the following steps.

**Step 1: PairingÂ **

The digits of the number are paired off, starting from the units digit and moving toward the left. Each pair is called period. Any single digit left at the end is also a period. The number of digits in the square root will be same as the number of periods.

Â

529 is paired as : two periods : 5 and 29

3969 is paired as : two periods :Â

44521 is paired as : three periods :

Step 2 : Division

Â

Long divisionÂ

Carefully see in the example shown for the method of long division. It is very different from the long division you have done till now. In the example shown, 529 is paired off as 5 and 29 (two periods).

(i) Find the largest number whose square is equal to or just less than the first period.

In this case this number is 2 since 2 Â´ 2 = 4.

Take this number as the divisor and quotient.

Â

(ii) Subtract the product from the first period. Bring down the next period. This is the new dividend. In this case, the new dividend is 129.

(iii) Multiply the quotient by 2 and write it as the next divisor. Remember to leave a blank space after this divisor.Â In this case the divisor is 2 Â´ 2 = 4, with a blank space after 4.

Â

(iv) Find the number (say x) which you can write in the blank space such that when the new number so formed is multiplied by x, the product is equal to or just less than the new dividend. Write x in the quotient and the product below the dividend.

Â

In this case we have 43 x 3 = 129 (i. e. x is 3) which is the same as the dividend. Write 3 in the quotient. Since the remainder is now 0, the square root is 23.

Â

Find the square root of (i) 6889Â Â (ii) 1046529

Â (i) 68 89

= 1023

Note : In (ii), after the first period you brought down the second period 04. There was no suitable number you could put down in the blank space after the divisor 2, therefore a '0' was put and the next period was brought down.

Â

The area of the square is 7744 m^{2}Â . Find its perimeter.

Area of square = (side)^{2}Â

.^{.}. Side =Â

Â i.e. Side = 88m

.^{.}. Perimeter = 4 x 88m = 352m

Â

What least number must be added to 7344 to make it a perfect square?

We go through the long division steps to find the square root of 7344. In the second division step

Â

we find thatÂ Â Â Â Â Â Â Â Â 165 x 5 = 825 ( < 944 )

whileÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 166 x 6 = 996 ( > 944 )

Â

Hence 7344 lies between 85^{2}Â and 86^{2}Â .

Â

To make it perfect square the least number that should be added is 86^{2}Â - 73 44 = 7396 - 7344 = 52

Â

What least number must be subtracted from 7344 to make it a perfect square

As we have seen in example above

85^{2Â }< 7344 < 86^{2}Â

To make 7344 into a perfect square, the least number that has to be subtracted isÂ

7344 - 85^{2}Â = 7344 - 7225 = 119