Square Root

Square root definition can be defined as a number which when multiplied by itself gives a as the product, then it is known as a square root. For example, since 

5 × 5 = 25, so \[\sqrt{25}\] = 5

All real numbers have two square roots, one is a positive square root and another one is a  negative square root. The positive square root is also referred to as the principal square root. Two positive numbers when multiplied results in a positive number because of having the same sign. As far as negative numbers are concerned, they don't have real square roots because a square is either positive or 0. Square root symbol is represented as  \[\sqrt{}\]

Perfect Squares

When two equal numbers are multiplied, it results in a perfect square. For example, 55=25

This is a basic property of the square root

Square Root Table

Referring to this square root table, half of the energy will be saved. 

How To Find Square Root?

Now, since we know what a square root is, we can quickly jump into knowing how to find a square root of a number. There is no square root formula as such but well, there are two ways to find the square root of a number. They are the Prime Factorization Method and the Division Method. You will know how to solve the square root equation using these two methods.

Finding Square Root Using Prime Factorization Method 

To find the square root of a perfect square we have to follow the following steps:

Step 1) First resolve the given number into prime factors.

Step 2) Make pairs of similar factors.

Step 3) The product of prime factors, chosen one out of every pair, gives the square root of the given number. 

For example, find the square root of  24336.

Solution) Resolving 24336 into prime factors, we get: 

24336 = \[\underline{2}\] × \[\underline{2}\] × \[\underline{2}\] ×  \[\underline{2}\] × \[\underline{3}\] ×  \[\underline{3}\]  × \[\underline{13}\]  × \[\underline{13}\]

\[\sqrt{24336}\] = 2 × 2 × 3 × 13 = 156

Finding Square Root Using Division Method

In case of large numbers; we find the square root by using the following steps: 

Step 1) Mark off the digits in pairs starting with the unit digit. Every pair and remaining one digit (if any) is called a period. 

Step 2) Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.

Step 3) In this step subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend. 

Step 4) now, a new divisor is obtained by taking twice 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 new divisor and this digit is equal to or just less than the new dividend.  

Solved Examples

Question 1) By using the table of square roots, find the values of 

i) \[\sqrt{13}\]

ii) \[\sqrt{83}\]

iii) \[\sqrt{150}\]

iv) \[\sqrt{36}\]

Solution 1)

i) Let x = 13

From the table, we find that when  x = 13, then \[\sqrt{x}\] 3.606.

Therefore, \[\sqrt{13}\] = 3.606.

ii) Let x = 83

From the table, we find that when  x = 83, then \[\sqrt{x}\] = 9.110.

Therefore,  \[\sqrt{83}\]=9.110.

iii) \[\sqrt{150}\] =  \[\sqrt{5 × 5 × 6}\] = 5 × \[\sqrt{6}\] = 5 × 2.449

Using the table for \[\sqrt{6}\],  = 12.245

Therefore, \[\sqrt{150}\] =  12.245

iv)\[\sqrt{368}\] = \[\sqrt{4 × 4 × 23}\] = 4 × \[\sqrt{23}\] = 4 × 4.796

Using the table for \[\sqrt{23}\], = 19.184

Therefore, \[\sqrt{368}\] = 19.184

Question 2) find the square root of  1764

Solution 2)

Resolving 1764 into prime factors, we get:

1764 = \[\underline{2}\] × \[\underline{2}\] × \[\underline{3}\] × \[\underline{3}\] × \[\underline{7}\] × \[\underline{7}\]

\[\sqrt{1764}\] = 2 × 3 × 7 = 42

Question 3) Find the smallest number which when multiplied by 720 gives a perfect square number. 

i) Give the perfect square number so obtained?

ii) Find the square root of this perfect square number.

Solution 3) Resolving 720 into prime factors, we get:

720 = 2 × 2  ×  2  ×  2  ×  3  ×  3  ×  5.

Thus, 2,2,3 exist in pairs while 5 is alone.

So, we should multiply the given number by 5 to get a perfect square number.

i) Perfect square number so obtained = 720 × 8 = 3600

ii) Now, 3600 = 2  ×  2  ×  2  ×  2  ×  3  ×  3  ×  5 ×  5

Therefore, \[\sqrt{3600}\] =   2 × 2  ×  3  ×  5 = 60