How to Find the Square Root: Step-by-Step Guide
Define a Square Root
A square root is a systemic function applied in the field of Mathematics. It is described as the value of a natural number written as y = \[\sqrt{a}\] . Here, the square root of ‘a’ is equal to ‘y’, where ‘a’ is a natural number. It can also be expressed as y\[^{2}\] = a. Therefore, the square root is defined as a number multiplied by itself would produce the original number. For example, 4 × 4 =16, and therefore the square root of 16 is 4. Square roots have been used since ancient times and provide a very simplistic way to solve numerical problems in algebra and geometry.
The Symbol of Square Root
The square root symbol is unique and well-known. It is represented as ‘\[\sqrt{}\]’ The symbol for a square root is also known as a radical and the number present under the root symbol is known as radicand.
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How to Find the Square Root?
To find the square root of a number, dependents on the type of data or number provided.
In the case of perfect square numbers, it is simple to obtain the original number by multiplying the prime number twice.
An example of the square root of \[\sqrt{4}\], we know that 2 × 2 = 4.
In the case of imperfect squares, a fraction is obtained instead. This may be a complicated process to follow, but, with a little practice, one can go a long way to solve problems related to this.
An example of the square root of 2\[\sqrt{2}\] = 1.414
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Types of Square Root Methods
A. Square Root by Prime Factorization
The prime factorization method to find the square root of a number is easy. Firstly, we factorization the number under the root sign followed by pairing them in two.
For example,
the square root of 9 is \[\sqrt{9}\] = \[\sqrt{3 \times 3}\] = 3
An easy way to remember how square roots work is that the inverse of squaring a number is finding its root. For example,
Hence, 12 = 1, the square root of 1 is 1
Hence, 42 = 16, square root of 16 is 4, and so on
When we take the prime factors of a number and its square,
For example 12 and 144,
12 = 2 х 2 х 3
144 = 2 х 2 х 3 х 2 х 2 х 3
Prime factorization of a square number provides prime factors that occur two times that of the number itself.
B. Square Root By Long Division
Below is an example, given to understand how long division works with the square root.
1. Let us take 484 as the number whose square root is to be evaluated. A bar is placed over the pair of numbers starting from the unit place. If there is a number, then a bar is placed over the left digit as well.
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2. The divisor is the largest number whose square is less than or equal to the number on the extreme left of the number. On dividing, write the quotient.
3. Here the quotient is 2 and the remainder is 0.
4. Next, the number with the bar is brought down to the right side of the remainder. Here, in this case, we bring down 84. Now, 84 is our new dividend.
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5. The value of the quotient is doubled and entered in the blank space on the right side.
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6. Select the largest digit for the unit place of the divisor (4_) such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than the dividend (84).
In this case, 42 × 2 = 84. So the new digit is 2.
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7. When the remainder is 0 there is no divisor left to divide, therefore, \[\sqrt{484}\] = 22
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FAQs on What Is Square Root? Meaning, Symbol & Examples
1. What is a square root in mathematics?
In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. It is the inverse operation of squaring a number. For example, the square root of 25 is 5, because 5 × 5 = 25. The symbol for square root is √, so we write it as √25 = 5.
2. What is the easiest method to find the square root of a perfect square?
The easiest method for finding the square root of a perfect square is the prime factorisation method. Follow these steps:
- First, break down the number into its prime factors.
- Next, group the identical factors into pairs.
- Finally, take one factor from each pair and multiply them together. The result is the square root.
For example, to find √144: Prime factors of 144 are (2 × 2) × (2 × 2) × (3 × 3). Taking one factor from each pair gives 2 × 2 × 3 = 12.
3. Where is the concept of a square root used in real-life examples?
The concept of square root has many real-world applications. Some important examples include:
- Architecture and Construction: To calculate the length of a diagonal or apply the Pythagorean theorem for right-angled structures.
- Geometry: To find the side length of a square if its area is known (Side = √Area).
- Physics: Used in formulas for calculating distance, speed, or gravitational forces.
- Finance: In calculating standard deviation to measure the volatility of investments.
4. Does a number have more than one square root?
Yes, every positive number has two square roots: one positive and one negative. For example, the square roots of 49 are +7 and -7, because both 7² (7x7) and (-7)² ((-7)x(-7)) equal 49. However, the radical symbol (√) specifically denotes the principal square root, which is the non-negative root. So, √49 is always considered to be 7.
5. What is the difference between finding the square of a number and its square root?
Finding the square and the square root are inverse operations.
- Squaring a number means multiplying the number by itself. For example, the square of 4 is 4² = 4 × 4 = 16.
- Finding the square root means finding which number, when multiplied by itself, gives the original number. For example, the square root of 16 is √16 = 4.
In short, one operation builds up a number, while the other breaks it down to its root.
6. How can we estimate the square root of a number that is not a perfect square, like 60?
To estimate the square root of a non-perfect square like 60, you can find the two perfect squares it lies between. We know that 7² = 49 and 8² = 64. Since 60 is between 49 and 64, its square root (√60) must be between 7 and 8. As 60 is closer to 64 than it is to 49, we can estimate that its square root will be closer to 8. A good estimate would be around 7.7. This method provides a quick approximation without a calculator.
7. What does a number in the form 'a√b' mean, for example, 3√2?
A number in the form 'a√b' represents 'a' multiplied by the square root of 'b'. So, 3√2 means 3 times the square root of 2. This is a way to express an irrational number precisely. You can also express it entirely under the radical sign by squaring the outside number and multiplying it inside: 3√2 = √(3² × 2) = √(9 × 2) = √18.
8. What are some key properties of square roots related to numbers ending in certain digits?
A key property is that numbers ending in 2, 3, 7, or 8 can never be perfect squares. This means you cannot find a whole number square root for any number with these digits in the unit's place. Additionally, the square root of an even perfect square is always an even number (e.g., √64 = 8), and the square root of an odd perfect square is always an odd number (e.g., √121 = 11).
