About Square Roots

To understand square roots we first need to know what is a square. If a number is multiplied by itself, the product so obtained is called the square of that number.
For example:1:  3 x 3 = 9. So we say that 9 is square of 3.
Example: 2: 16 x 16 = 256. Then we say that 256 is the square of 16.

Square root definition:

The number when multiplied by itself gives its square is called square root of the other number. In other words, if a x a = b, then we say that a is the square root of b.
For example: 1: 3 is the square root of 9, because 3 x 3 = 9 or 3^2 = 9
Example:2: 11 is the square root of 121, again because 11^2 = 121

How to find square roots (or how to do square roots):

There are primarily two methods for finding square root:

Method 1: By prime factorization
Step 1: Resolve the given number into prime factors.
Step 2: Form pairs of like factors
Step 3: From each pair, pick out one prime factor
Step 4: Multiply the factors so picked.

Example:1: Find square root of 7225
Step 1: Resolving into prime factors and forming pairs of like factors
7225 = 5 x 5 x 17 x 17
Step 2: From each pair pick out one prime factor, i.e.,
From first pair we pick 5 and from second pair we pick 17.
Step 3: Multiply the factors so picked. So 5 x 17 = 85.
Answer: square root of 7225 = 85 or √7225 = 85

Method 2: By long division
Example 1: Calculate square root of 17424
Step 1: Mark off the digits in pairs from right to left
Step 2: 1^2=1
Step 3: Twice 1 = 2
Step 4: 2 goes into 7 three times. Put 3 on top and in the divisor as shown 23 x 3 = 69
Step 5: Double 13 = 26. First digit of 26 is 2, goes into the first digit of 524 which is 5, 2 times. Place 2 on top and in divisor as shown. 2 x 262 = 524
Step 6: Subtract. Remainder is zero and square root is 132.
Answer: √17424 = 132