Square Root Prime Factorization Explained

Have you ever wondered why students are a little uneasy with the subject of Mathematics? Is the subject actually difficult to understand and study? Is it really hard to score good marks in math? Will a student ever fall in love with the subject?

The subject experts at Vedantu patiently have studied the challenges of students across their network and have made the following conclusions - 

• Students find a subject difficult when they fail to understand the basics clearly

• The teaching method and learning techniques also plays a role in building a student’s relationship with a subject 

• The guidance and the reading materials made available to the students also decide their interest in the subject

Considering all these problems, our team tries their best to make reading more interesting and fun for the students. Students can download the free reading materials where concepts are explained in the easiest language. Video lectures are made available to make students understand better. 

This particular article brings another mathematical concept, explained in detail for the students to grasp the concept and get familiar with mathematical concepts.

Table of content - 

• Square Root - Introduction

• Square Root Definition

• Method of finding the square root

• Prime factorization method

• Solved examples 

• Fun facts 

• Frequently asked questions

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 A^1/2. 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 as follows.

So, 576 can be written as a product of prime numbers as: 576=2×2×2×2×2×2×3×3

Square root of 576 = 2×2×2×3=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.

1764=2×2×3×3×7×7

Step 2:

Identical factors are paired. 

1764=2×2×3×3×7×7

Step 3: One factor from each pair is chosen and the product is found to get the square root. √1764=2×3×7

                    √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:

11025=3×3×5×5×7×7

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.

√11025=3×5×7=105

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

Solution:

Using the prime factorization method, 8712 can be factorized as

8712=2×2×2×3×3×11×11

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

8712=2×2×2×3×3×11×11

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 the number ‘x’ can be written as ±√x.

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

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