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Finding Square Roots through the Prime Factorization is one of the efficient methods to find the answers to the related questions. A square root is a result that generates the same number which is multiplied by itself. In the prime factorization method, the factors of the number under the root are taken and then grouped in a pair of two. For example, the square root of 16 is = =4. This is when we find the square root of a perfect square. However, to calculate the square root of an imperfect square, we tend to perform the long division method to calculate it.
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Square Root
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The square of a number is the number that is multiplied by itself to generate the square. The symbol which is used to denote the square root is √. It can also be replaced by the exponent or the power of ½. Thus, we could say that if we have a number x, then the square root of x can be denoted as or x½. Every number has two roots that have the same magnitude but opposite signs. They can be used to estimate a side of the square when the area is given.
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Ways to Find Square Root
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Several methods can be considered while calculating the square root of any number. Some of the ways to calculate the square root are given below:
- Prime Factorization Method
- Number Line Method
- Guess and Check Method
- Repeated Subtraction Method
- Long Division Method
- Average Method
While the prime factorization and repeated subtraction methods are used to calculate the square root of the perfect squares only. The perfect square refers to the numbers which have the square root as integers. For example, the perfect square numbers can be 1, 4, 9, 16, 25, 36, 49, and so on.
Read More: Area of Rectangle
Prime Factorization Method
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Prime Factorization Method is used to formulate the square root of a perfect square only. Prime factorization involves the method of factoring any particular number into its prime factors and then grouping it into pairs. The pairs are then taken as a single unit. It also helps us in determining whether a specific number is a perfect square. It can, however, not calculatethe square root of any decimal number which does not come under the perfect square. They can be used in encryption, cryptography, decoding the number-based street locks and by computer specialists, manufacturers, and product designers.
Example: Calculate the square root of 576.
Solution: Firstly, factoring the 576 into its prime factors:
| 2 | 576 |
| 2 | 288 |
| 2 | 144 |
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
The prime factors of 576 are: 2, 2, 2, 2, 2, 3, 3
We can write 576= 2×2×2×2×2×3×3
The square root of 576=2×2×2×3=24
Therefore, 576 = 24.
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Solved Examples
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Some examples of finding out square root using prime factorization method are given below:
Example1. Estimate the square root of 1764 with the help of the prime factorization method.
Solution1. Finding out the prime factors of 1764, we have:
| 2 | 1764 |
| 2 | 882 |
| 3 | 441 |
| 3 | 147 |
| 7 | 49 |
| 7 | 7 |
| 1 |
Now, pairing the prime factors:
1764=2×2×3×3×7×7
Now, taking the pair as a single unit and then finally calculating the square root, we have:
1764 = 2×3×7= 42
1764 = 42
Example2. Find out if 11025 is a perfect square or not. Find the square root with the help of the prime factorization method if the number is a perfect square.
Solution2. The prime factors of 11025 using the prime factorization method are as follows:
| 3 | 11025 |
| 3 | 3675 |
| 5 | 1225 |
| 5 | 246 |
| 7 | 49 |
| 7 | 7 |
| 1 |
110253=3×3×5×5×7×7
As we can see that all the prime factors can be grouped into pairs, we have no prime factor left. Therefore, 11025 is a perfect square.
11025 = 3×5×7
11025 = 105
Example3. Estimate the minimum number to be multiplied by 8712 so that it becomes a perfect square.
Solution3. Calculating the prime factors of 8712 we have,
| 2 | 8712 |
| 2 | 4356 |
| 2 | 2178 |
| 3 | 1089 |
| 3 | 363 |
| 11 | 121 |
| 11 | 11 |
| 1 |
8712= 2×2×2×3×3×11×11
Pairing these prime factors, we get:
8712= 2×2×2×3×3×11×11
As we know, 2 is left unpaired, therefore, we need to multiply 8712 by 2 to make it a perfect square.
Check Also: Conjunction
Things to Remember
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- When any number gets multiplied by itself, the number which is multiplied is called a square root.
- It can be stated in the form of power of ½ or can be denoted by √.
- The roots tend to have equal magnitudes and opposite signs.
- Several ways can be used to find the square root of any number.
- Prime factorization involves the method of estimating the square root of a number by breaking it into prime factors and pairing it.
- The prime factorization and repeated subtraction are used only for the numbers that are perfect squares.
- The perfect squares are the numbers that have integers such as 1, 4, 9, and so on.
Sample Questions
Ques1. Find out if 120 is a perfect square or not using the prime factorization method.
Ans1. The prime factors of 120 using the prime factorization method are as follows:
| 2 | 120 |
| 2 | 60 |
| 2 | 30 |
| 3 | 15 |
| 5 | 5 |
| 1 |
120=2×2×2×3×5
As we can see the prime factors that are 2,3, and 5 cannot be grouped into pairs. Therefore, 120 is not a perfect square.
Ques2. Calculate the square root of 1024.
Ans2. The square root of 1024 is as:
| 2 | 1024 |
| 2 | 512 |
| 2 | 256 |
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 2 | 1 |
Therefore, 1024= 2×2×2×2×2×2×2×2×2×2
1024 = 2×2×2×2×2= 32
Ques3. Calculate the square root of 324.
Ans3. The square root of 324 is as:
| 2 | 324 |
| 2 | 162 |
| 2 | 81 |
| 3 | 27 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Therefore, 324= 2×2×2×3×3×3
324 = 2×3×3= 18
Ques4. Calculate the square root of 2025.
Ans4. The square root of 2025 is as:
| 3 | 2025 |
| 3 | 675 |
| 3 | 225 |
| 3 | 75 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Therefore, 2025= 3×3×3×3×5×5
2025 = 3×3×5= 45
Ques5. Calculate the square root of 4096.
Ans5. The square root of 4096 is as:
| 2 | 4096 |
| 2 | 2048 |
| 2 | 1024 |
| 2 | 512 |
| 2 | 256 |
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Therefore, 4096= 2×2×2×2×2×2×2×2×2×2×2×2
4096 = 2×2×2×2×2×2= 64
Ques6. Calculate the square root of 400.
Ans6. The square root of 400 is as:
| 2 | 400 |
| 2 | 200 |
| 2 | 100 |
| 2 | 50 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Therefore, 400= 2×2×2×2×5×5
400 = 2×2×5= 20
Ques7. Calculate the square root of 81.
Ans7. The square root of 81 is as:
| 3 | 81 |
| 3 | 27 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Therefore, 81= 3×3×3×3
81 = 3×3= 9
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