Square Root Prime Factorization

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Square Root Basics

We all are aware of a geometrical shape, the square. Square is a geometrical shape which has four sides of equal length and angles equal to 900. Square, being a two dimensional shape, covers a specific surface of the plane. This region covered by the square is called its area. Area of a square is calculated as the side x side. If the area of the square is given and its side is to be determined then we use an operation in Mathematics called the square root. For example, if the area of a square is 9 sq. units, then its side measures 3 units which is calculated as the square root of 9.


Square Root Definition

Square of a number is another number obtained by multiplying the number by itself. Square root is the inverse operation of square. Square root of a number is that number which when multiplied by itself, gives the number whose square root is to be determined as the answer. For example, when 7 is multiplied by itself, the product obtained is 49. Therefore, we can say that the square root of 49 is 7. Square root of a number is represented by the symbol ‘√’. It can also be represented exponentially as the number to the power ½ . The square root of a number ‘A’ can be represented as √A or A1/2. Any number in Mathematics will have two roots of equal magnitude and opposite sign.


Methods to Find Square Root of a Number

Square root of a number can be determined by various methods. A few popular methods used to find the square root of a number are:

  1. Guess and check Method.

  2. Average Method.

  3. Repeated Subtraction Method.

  4. Prime Factorization method.

  5. Long Division Method.

  6. Number Line Method.

The repeated subtraction method and prime factorization method is applicable only for perfect square numbers. Perfect square numbers are the numbers whose square roots are integers. The examples for perfect square numbers are 1, 4, 9, 16, 25 …… 


How to Find the Square Root of a Number by Prime Factorization Method?

Prime factorization method is a method in which the numbers are expressed as a product of their prime factors. The identical prime factors are paired and the product of one element from each pair gives the square root of the number. This method can also be used to find whether a number is a perfect square or not. However, this method cannot be used to find the square root of decimal numbers which are not perfect squares.


Example: Evaluate the root of 576.


Solution: 

576 is factorized into its prime factors are follows.


2

576

2

288

2

144

2

72

2

36

2

18

3

9

3

3


1


So, 576 can be written as a product of prime numbers as:


\[576 = 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\]


Square root of 576 = \[2\times 2\times 2\times 3 = 24\]


Square Root by Prime Factorization Example Problems

1. Find the square root of 1764 using the prime factorization method.


Solution:

Step 1:

The given number is resolved into its prime factors.


2

1764

2

882

3

441

3

147

7

49


7


\[1764 = 2\times 2\times 3\times 3\times 7\times 7\]


Step 2:

Identical factors are paired.


\[1764 = 2\times 2\times 3\times 3\times 7\times 7\]


Step 3:

One factor from each pair is chosen and the product is found to get the square root. 


\[\sqrt{1764} = 2 \times 3 \times 7\]


\[\sqrt{1764} =  42\]


2. Check whether 11025 is a perfect square or not. If it is a perfect square, find its square root by factorization method. 


Solution:

Using prime factorization method, 11025 can be written as the product of its primes as:


3

11025

3

3675

5

1225

5

245

7

49


7


\[11025 = 3\times 3\times 5\times 5\times 7\times 7\]


All the prime factors can be grouped into pairs of identical factors. No prime factor is left all alone. Hence 11025 is a perfect square number.


\[\sqrt{11025} = 3 \times 5 \times 7 = 105\]


3. Find the smallest number to be multiplied by 8712 to make it a perfect square number.


Solution:

Using prime factorization method, 8712 can be factorized as:


2

8712

2

4356

2

2178

3

1089

3

363

11

121


11


\[8712 = 2 \times 2 \times 2 \times 3\times 3\times 11\times 11\]


When the identical factors are paired, 8712 can be written as:


\[8712 = 2\times 2\times 2\times 3\times 3\times 11\times 11\]


So, the number 8712 should be multiplied by 2 in order to get a perfect square number. 


Fun Facts:

  • Any real number has two square roots: a positive root and a negative root. Both the roots are the same in magnitude but the signs are opposite. So, the square root of a number ‘x’ can be written as ±√x.

  • Square root of a square of any number is the number itself.

  • Square root of non perfect square numbers cannot be determined using prime factorization method. However, one can determine the number to be multiplied or divided by the given number to make it a perfect square.

FAQ (Frequently Asked Questions)

1. What are the Properties of the Square Root of Decimals Which are Perfect Squares?

Answer:

  • The natural numbers ending with 2, 3, 7 and 8 do not have perfect square roots. Only the numbers ending with 0, 1, 4, 5, 6 and 9 may have square roots.

  • Numbers ending with an odd number of zeros do not square roots. Numbers ending with an even number of zeros have half the number of zeros in their square root. 

  • Square root of an even square number is even and that of an odd square number is odd. 

  • Every number has two roots: One positive root and one negative root.

  • Square roots of negative numbers are imaginary. 

2. How to Find the Square Root of a Number by Repeated Subtraction Method?

Answer: This is a method in which the number whose square root is to be determined is repeatedly subtracted by the consecutive odd number till the difference becomes zero. The number of subtractions give the root of the number.This method can only be used to find the square root of perfect square numbers.


Example: Estimate the square root of 9


Solution:

The number is subtracted from odd numbers starting from 1.

9 - 1 = 8

8 - 3 = 5

5 - 5 = 0

Number of subtractions here is 3. So, the square root of 9 is 3.