SQUARE ROOTS

About "Square roots"

Square roots :

The square root of a number is the value such that, when a number multiplied by itself,for example

   3 x 3 = 9


It is written with a radical symbol " √ " and the number or expression inside the radical symbol is called the radicand.

Methods to find square-root of perfect square

(i) Prime factorization 

(ii) Long division method

How to find the square-root of a number using prime factorization

  • Using this method, first we have to split the given number into prime factors.
  • Write those prime factors inside the radical sign instead of the given number.
  • Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.
  • Multiply the numbers which have come out from the radical sign.

Examples

Question 1 : 

Find square-root of 324 by prime factorization

Solution :

Step 1 :

Split 324 into prime factors 

Step 2 :

324  =  √2 x 2 x 3 x 3 x 3 x 3 

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

Hence, square root of 324 is 18.

Question 2 : 

Find square root of 625 by prime factorization

Solution :

Step 1 :

Split 625 into prime factors 

Step 2 :

√625  =  √5 x 5 x 5 x 5

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

  =  5 x 5

  =  25

Hence, √625 is 25.

Question 3 : 

Find square-root of 4096 by prime factorization

Solution :

Step 1 :

Split 625 into prime factors 

Step 2 :

√4096  =  √2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Step 3 :

Take one common number from the radical

  =  2 x 2 x 2 x 2

  =  64

Hence, square-root of 4096 is 64.

Question 4 : 

Find square-root of 400 by prime factorization

Solution :

Step 1 :

Split 400 into prime factors 

Step 2 :

√400  =  √2 x 2 x 2 x 2 x 5 x 5

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

  =  2 x 2 x 5

  =  20

Hence, square root of √400 is 20.

Question 5 : 

Find square-root of 144 by prime factorization

Solution :

Step 1 :

Split 144 into prime factors 

Step 2 :

√144  =  √2 x 2 x 2 x 2 x 3 x 3

Step 3 :

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

  =  2 x 2 x 3

  =  12

Hence, √144  is 12.

Question 6 : 

Find the square-root of 104976 by long division method  

Solution :

Step 1 :

Separate the digits by taking commas from right to left once in two digits.

10,49,76  

When we do so, we get 10 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product ≤ 10

(The product must be greatest and also less than 10)

The above condition will be met by “3”.

Because 3x3 = 9 ≤ 10

Now this situation is explained using long division

In the above picture, 9 is subtracted from 10 and we got the remainder 1.

Step 3 :

Now, we have to bring down 49 and quotient 3 to be multiplied by 2 as given in the picture below.

Step 4 :

Now we have to take a same number at the two places indicated by "?".

Then, we have to find the product as shown in the picture and also the product must meet the condition as indicated.

Step 5 :

The condition said in step 4 will be met by replacing "?" with "2". 

Than we have to do the calculation as given in the picture. 

Step 6 :

Now, we have to bring down 76 and quotient 32 to be multiplied by 2 as given in the picture below.

Step 7 :

In the above picture, we have applied the procedures explained in step 4 and step 5. And we got the remainder zero.  

Step 8 :

From the above picture, finally we got the square root of 104976. That is 324.

Hence, the square root of 104976 = 324

After having gone through the stuff given above, we hope that the students would have understood "Square-roots"

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