SQUARE ROOTS OF PERFECT SQUARE

About "Finding square roots of perfect squares"

Finding square roots of perfect squares :

Finding square roots is the opposite process squaring. 

1²  =  1, the square root of 1 is 1.

2²  =  4, the square root of 4 is 2.

3²  =  9, the square root of 9 is 3.

4²  =  16, the square root of 16 is 4.

what is perfect square ?

From the above example, we come to know that 16, 9, 4 and 1 are known as perfect squares. Because we can represent all as the multiples two same numbers.

That is 16 = 4 x 4, 9 = 3 x 3, 4 = 2 x 2, 1 = 1 x 1.

Like wise 18 is not a perfect square, because we cannot represent this 18 as the multiple of two same terms.  

Usually we follow two methods to find square roots of perfect squares.

• Prime factorization method
• Long division method

Let us see some example problems to understand the method of finding square root using the above methods.

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.

Question 1 :

Find the prime factors of 324

Solution :

Step 1 :

Since the given number ends with 4, first we have to split the given number by the smallest even prime number 2.

Step 2 :

2 goes into 3 one time.We have 1 left. If we take this 1 along with the next digit 2, we get 12. If we divide this by 2, we get 6. 

We don’t have any number remaining in 12. So we can take the next digit 4. Again, if we divide 4 by 2, we get 2.

Step 3 :

√324 = √(2 x 2 x 3 x 3 x 3 x 3)

  =  2 x 3 x 3

  =  18

Hence the square square root of 324 is 18.

We can do the same problem using long division method

Step 1 :

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

3, 24

When we do so, we get 3 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product ≤ 3

(The product must be greatest and also less than 3)

The above condition will be met by “1”.

Because 1 x 1 = 1 ≤ 3, but 2 x 2 = 4 > 3

Now this situation is explained using long division

In the above picture, 1 is subtracted from 3 and we got the remainder 2.

Step 3 :

Now, we have to bring down 24 and quotient 1 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. 

Hence the square square root of 324 is 18.

Question 2 :

Find the prime factors of 625

Solution :

Step 1 :

Since the given number ends with 5, first we have to split it by the prime number 5.

Step 2 :

5 goes into 6 one time.We have 1 left. If we take this 1 along with the next digit 2, we get 12. Again we have to divide it by 5. If we divide this by 5, we get 2.

Now we have 2 left. Now we have to take this 2 along with the next digit 5, we get 25.  If we divide 25 by 5, we get 5.

Step 3 :

By repeating this process until we get prime factors. 

Hence, √625 = √(5 x 5 x 5 x 5)  =  5 x 5  = 25

We can do the same problem using long division method

Step 1 :

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

3, 24

When we do so, we get 3 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product ≤ 6

(The product must be greatest and also less than 6)

The above condition will be met by “1”.

Because 2 x 2 = 4 ≤ 6, but 3 x 3 = 9 > 6

Now this situation is explained using long division

In the above picture, 4 is subtracted from 6 and we got the remainder 2.

Step 3 :

Now, we have to bring down 25 and quotient 1 to be multiplied by 2.

Now we have put a number next to 4, and the same number should be in the quotient next to 2. So that their product must be less than or equal to 225.

By multiplying 5 and 45, we will get the answer 225.

Hence 25 is the square root of 625.

After having gone through the stuff given above, we hope that the students would have understood "Square roots of perfect squares". 

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