Interested in learning about the square root method? Keep reading to learn what it is, and how to use it effectively.
Using square roots in mathematics is limitless. Students must know what the square root method is. The square root of a number is found when the original number is multiplied by itself. We’ll break down how to find a square root, which methods to use, and go through several examples to make sure you understand how to put them into practice.
The first thing you need to know to find the square root of a number is if the number is a perfect or imperfect square. A number with an imperfect square where the square root gives a fractional or decimal value. For example, perfect squares include 4, 16, and 64, while examples of imperfect squares are 5, 7, 19, and 23.
Once you know if the number is a perfect or imperfect square, you can figure out the method for finding the square root. Here are some ways you can find the square root of a number:
It’s easy to calculate the square root of a number using the prime factorization method. The prime factorization of a number is the product of the prime numbers. You can only divide prime numbers by 1 or the number itself. Below, we list the steps you need to take when using this method to determine the square root of a given number:
Here are some more examples to illustrate the steps above:
| Number | Prime Factorization | Square Root |
|---|---|---|
| 4 | 2 x 2 | √4 = 2 |
| 9 | (3 x 3) | √9 = 3 |
| 16 | (2 x 2) x (2 x 2) | √16 = 2 x 2 = 4 |
| 144 | (2 x 2) x (2 x 2) x (3 x 3) | √144 = 2 x 2 x 3 = 12 |
| 169 | (2 x 2) x (2 x 2) x (3 x 3) | √169 = 13 |
| 256 | (13 x 13) | √256 = 2 x 2 x 2 x 2 = 16 |
| 324 | (2 x 2) x (3 x 3) x (3 x 3) | √324 = 2 x 3 x 3 = 18 |
This method works well when the number given is a perfect square. Otherwise, we will need to use other methods.
This is the simplest method. You can get the square root of a given number by subtracting consecutive odd numbers from the number given until it reaches 0. Hence, the number of times the subtraction takes place is the square root of the given number. Like the method above, this also works for perfect square numbers only. Below are some examples:
Answer and Explanation
The number of times that we subtracted from 16 is four. Hence, √16 = 4
Answer and Explanation
The number of times we subtracted from 25 is five times. Hence √25 = 5
Answer and Explanation
9 - 1 = 8
8 - 3 = 5
5 - 5 = 0
The number of times we subtracted from 9 is three times. Hence √9 = 3
Answer and Explanation
4 - 1 = 3
3 - 3 = 0
The number of times we subtracted from 4 is two times. Hence √4=2
Answer and Explanation
64 - 1 = 63
63 - 3 = 60
60 - 5 = 55
55 - 7 = 48
48 - 9 = 39
39 - 11 = 28
28 - 13 = 15
15 - 15 = 0
The number of times we subtracted from 64 is eight times. Hence √64 = 8
The long division method is suitable for numbers that are not perfect square numbers. It divides large numbers into steps or parts, which breaks the number into sequences. This method can give you the exact square root of the number given. Below are the steps using 164 as the given value:
Step 1: Place a bar over the two digits that we have, i.e., 6 and 4
Step 2: Divide the left-most number by the most significant number whose square is less than or equal to the number in the left-most pair.
Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. Then, to the right of the obtained sum, find a suitable number, which, together with the result of the sum, forms a new divisor for the new dividend carried down. The terminologies are explained below:
Step 4: The new number in the quotient will have the same number as the divisor, which can be either less or equal to the dividend.
Step 5: The process continues using a decimal point and adding zeros in pairs underneath the remainder.
Hence, the quotient obtained is the square of the given number. Hence, the square root of 64 is 8. Below are some of the examples:
1. √64 = 8
2. √169 = 13
3. √144 = 12
4. √400 = 20
5. √1024 = 32
This method is long, time-consuming, and has to deal with guesses. People who are familiar with finding square roots may find the method straightforward. By definition, the estimation method is a reasonable guess that can give the actual value. It can also be used for values that cannot give perfect squares. Below are some examples with their explanation:
1. Find the square root of 15.
Answers and Explanation
The closest perfect squares to 15 are 16 and 9. The square roots of both numbers are 4 and 3, respectively. Therefore, it’s a good guess to say the √15 lies between 3 and 4.
If we take our guesses further, √15 will be closer to 4 than 3 since 16 is closer to 15 than 9. Thus, the square root will be greater than 3.5. Furthermore, the square of 3.8 is 14.44, and that of 3.9 is 15.21.
So, the √15 is between 3.8 and 3.9. Another guess demonstrates that the √15 is 3.87
2. Find the square root of 17.
Answer and Explanation
The closest perfect squares to 17 are 16 and 25, with their square root of 4 and 5, respectively. The √17 should be around 4. Since 17 is closer to 16 than 25. Therefore, the √17 is 4.123.
3. Find the square root of 19.
Answer and Explanation
The closest perfect squares to 19 are 16 and 25, with their square root of 4 and 5, respectively. So, the √19 will be between 4 and 5. 19 is closer to 16 than 25. So, the answer will be less than 4.5. More guesses demonstrate that the answer is 4.358.
4. Find the square root of 29.
Answer and Explanation
The closest perfect squares to 29 are 25 and 36, with their square root of 5 and 6, respectively. So, the √29 will be between 5 and 6. 29 is closer to 25 than 36. Thus, the answer will be less than 5.5. If we take our guesses further, we find that the answer is 5.385.
5. Find the square root of 90.
Answer and Explanation
The closest perfect squares to 90 are 81 and 100, with their square root of 9 and 10, respectively. So, the √90 will be between 81 and 100. 90 is nearly between 81 and 100 but is closer to 81. Thus, the answer will be close to 9.5. If we guess further, we find that the answer is 9.48.
Exams taken in schools often have questions on the square root of numbers. Below are some examples of the exam questions:
1. Find the square root of the 7744 by prime factorization
Answer and Explanation
√7744 = √ [(2 × 2) × (2 × 2) ×(2 × 2) × (11 × 11)] = 2 × 2 × 2 × 11 = 88
2. Find the square root of the 1156 by prime factorization
Answer and Explanation
√1156 = √ [(2 x 2) × (17 x 17)] = 2 x 17 = 34
3. Find the square root of 81 by prime factorization
Answer and Explanation
√81 = √ (3 x 3) x (3 x 3)] = 3 x 3 = 9
4. Find the √144 using the repeated subtraction method
Answer and Explanation
144 - 1 = 143
143 - 3 = 140
140 - 5 = 135
135 - 7 = 128
128 - 9 = 119
119 - 11 = 108
108 - 13 = 95
95 - 15 = 80
80 - 17 = 63
63 - 19 = 44
44 - 21 = 23
23 - 23 = 0
Hence, the √144 is 12 because we subtracted 12 times before reaching zero.
5. Find the square root of 24 using the estimation method.
The closest perfect squares to 24 are 16 and 25, respectively, with their square root of 4 and 15, Hence, the √24 will be between 4 and 5. The 24 is closer to 25 than the 16. Hence, the answer will be greater than 4.5 and be closer to 5. Further guesses show the answer is 4.89.
Do you still have more questions about finding the square root of a given number? Below are some frequently asked questions and answers:
The square root of a number is the value that gives the initial number when multiplied by itself.
You can find a square root using either the prime factorization, long division, repeated subtraction, or estimation method.
Squaring a number means multiplying it by itself. The square root of a number is the value that, when multiplied by itself, equals the original number.
To find the square root of a perfect square, just look for the number that, when multiplied by itself, gives you the perfect square. For example, to find the square root of 9, ask yourself, “What number times itself equals 9?” The answer is 3 because 3 x 3 = 9.
To find the square root of imperfect squares, you’ll need to use the estimation method.
The square root has many practical applications in mathematics and computing, such as solving quadratic equations, simplifying algebraic expressions, and calculating areas in geometry.
There are different ways to find the square root of a number. With the explanation above, you can correctly answer any questions when finding a number's square root using different methods. You will also be able to answer the question, "What is the square root method, and how do I use it?"
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