Square Root Definition Formula Properties and Solved Examples
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 in Mathematics
1. What is a square root in Maths?
A square root of a number is a value that, when multiplied by itself, gives the original number. In simple terms, if x × x = n, then x is the square root of n.
- Example: Since 5 × 5 = 25, the square root of 25 is 5.
- The symbol used for square root is √.
- Every positive number has two square roots: one positive and one negative (e.g., ±5 for 25).
2. How do you find the square root of a number?
You can find the square root by factorization, prime factorization, estimation, or using a calculator. Common methods include:
- Prime factorization method: Break the number into prime factors and pair them.
- Long division method: Used for large or non-perfect squares.
- Calculator method: Press the √ button.
3. What is the square root formula?
The basic square root formula is written as √x = x1/2, which represents a number raised to the power of one-half. Related formulas include:
- √(a × b) = √a × √b
- √(a / b) = √a / √b (b ≠ 0)
- (√a)2 = a
4. What is the square root of 0?
The square root of 0 is 0 because 0 multiplied by itself equals 0. In other words:
- 0 × 0 = 0
- Therefore, √0 = 0
5. What is the square root of a negative number?
The square root of a negative number is not a real number but an imaginary number. In mathematics:
- √(-1) = i, where i is the imaginary unit.
- For example, √(-9) = 3i.
6. What is a perfect square?
A perfect square is a number that is the square of an integer. This means it has a whole number as its square root.
- Examples: 1, 4, 9, 16, 25
- √16 = 4
- √25 = 5
7. What is the difference between square and square root?
The square of a number is obtained by multiplying it by itself, while the square root is the reverse process.
- Square: 7² = 7 × 7 = 49
- Square root: √49 = 7
8. How do you simplify a square root?
To simplify a square root, factor the number into perfect squares and take them outside the root symbol. Steps:
- Factor the number.
- Identify perfect square factors.
- Simplify using √(a × b) = √a × √b.
9. Why do we use square roots in real life?
Square roots are used to find unknown lengths, distances, and values in geometry, physics, and engineering. Common applications include:
- Pythagoras' theorem to calculate triangle sides.
- Finding distance using the distance formula.
- Calculating standard deviation in statistics.
10. Can you give an example of finding a square root?
Yes, to find the square root of 49, we look for a number that multiplies by itself to give 49.
- 7 × 7 = 49
- Therefore, √49 = 7
