How to Find Square Root Using Prime Factorization Steps and Examples
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 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:
Guess and Check Method.
Average Method.
Repeated Subtraction Method.
Prime Factorization method.
Long Division Method.
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.
FAQs on Square Root Prime Factorization Method
1. What is square root prime factorization?
Square root prime factorization is a method of finding the square root of a number by first expressing it as a product of prime factors and then grouping them in pairs. The steps are:
- Write the number as a product of prime numbers.
- Group identical prime factors in pairs.
- Take one factor from each pair outside the square root.
This method is especially useful for finding the square root of large perfect squares.
2. How do you find the square root using prime factorization?
To find the square root using prime factorization, factor the number into primes and take one number from each pair of identical factors. Follow these steps:
- Example: Find √36.
- Prime factorization: 36 = 2 × 2 × 3 × 3.
- Group into pairs: (2 × 2) and (3 × 3).
- Take one from each pair: √36 = 2 × 3 = 6.
This works because √(a²) = a for any positive number a.
3. What is the prime factorization of a number?
Prime factorization is the process of expressing a number as a product of its prime numbers only. A prime number has exactly two factors: 1 and itself.
- Example: 24 = 2 × 2 × 2 × 3.
- This can also be written as 2³ × 3.
Prime factorization is the foundation for finding square roots, HCF, and LCM.
4. Why do we group prime factors in pairs when finding square roots?
We group prime factors in pairs because a square root means finding two identical factors multiplied together. Since √(a²) = a, each pair of identical primes forms a perfect square.
- Example: √(2 × 2) = √(2²) = 2.
- If a factor has no pair, it remains inside the square root.
Pairing makes it easy to simplify square roots accurately.
5. Can you give an example of finding the square root of 144 using prime factorization?
The square root of 144 using prime factorization is 12. Here is the process:
- Prime factorization: 144 = 2 × 2 × 2 × 2 × 3 × 3.
- Group pairs: (2 × 2), (2 × 2), (3 × 3).
- Take one from each pair: 2 × 2 × 3 = 12.
Thus, √144 = 12 because 12 × 12 = 144.
6. How do you simplify a non-perfect square using prime factorization?
To simplify a non-perfect square, use prime factorization and take out only complete pairs of factors. For example:
- Simplify √72.
- Prime factorization: 72 = 2 × 2 × 2 × 3 × 3.
- Group pairs: (2 × 2), (3 × 3), and one 2 left.
- Result: √72 = 6√2.
The unpaired factor remains inside the square root.
7. What is a perfect square in prime factorization?
A perfect square is a number whose prime factors all appear in even powers. This means every prime factor has a pair.
- Example: 100 = 2² × 5².
- Since all exponents are even, √100 = 10.
If any prime factor has an odd power, the number is not a perfect square.
8. What happens if a prime factor has no pair when finding a square root?
If a prime factor has no pair, it stays inside the square root sign as part of the simplified radical. Only complete pairs can be taken outside.
- Example: √18.
- 18 = 2 × 3 × 3.
- Pair: (3 × 3), leftover: 2.
- Result: √18 = 3√2.
Unpaired factors prevent the number from being a perfect square.
9. Is prime factorization method better than long division method for square roots?
The prime factorization method is simpler for perfect squares, while the long division method is better for finding square roots of non-perfect squares to decimals. Comparison:
- Prime factorization: Clear and accurate for exact roots.
- Long division method: Useful for approximations.
Both methods are important in learning square root calculations.
10. What are common mistakes in square root prime factorization?
Common mistakes in square root prime factorization include incorrect prime factoring and missing pairs of factors. Avoid these errors:
- Stopping factorization before reaching prime numbers.
- Forgetting to group identical factors correctly.
- Taking out unpaired factors from the square root.
- Arithmetic multiplication mistakes.
Always check that all factors are prime and grouped properly before writing the final answer.
