Square root of a number is a value, which on multiplied by itself gives the original number. Suppose, x is the square root of y, then it is represented as x=√y or we can express the same equation as x^{2} = y. Here,’√’is the radical symbol used to represent the root of numbers. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number.
For example, the square of 3 is 9, 3^{2} = 9 and the square root of 9, √9 = 3. Since 9 is a perfect square, hence it is easy to find the square root of such numbers, but for an imperfect square like 3, 7, 5, etc., it is really tricky to find the root. Learn square roots from 1 to 25 with some shortcut tricks here.
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Definition
The square root of any number is equal to a number, which when squared gives the original number.
Let us say m is a positive integer, such that √(m.m) = √(m^{2}) = m
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Note: The square root of a negative number represents a complex number.
Suppose √-n = i√n, where i is the imaginary number.
Symbol
The square root symbol is usually denoted as ‘√’. It is called a radical symbol. To represent a number ‘x’ as a square root using this symbol can be written as:
‘ √x ‘
where x is the number. The number under the radical symbol is called the radicand. For example, the square root of 6 is also represented as radical of 6. Both represent the same value.
Formula
The formula to find the square root is:
y = √a |
Since, y.y = y^{2} = a; where ‘a’ is the square of a number ‘y’.
Properties
In mathematics, a square root function is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.
f(x) = √x
For example, if x=9, then the function returns the output value as 3. Some of the important properties of the square root are as follows:
- If a number is a perfect square number, then there exists a perfect square root.
- If a number ends with even number of zeros (0’s), then it can have a square root.
- The two square root values can be multiplied. For example, √3 can be multiplied by √2, then the result should be √6.
- When two same square roots are multiplied, then the result should be a radicand number. It means that the result is a non-square root number. For instance, when √7 is multiplied by √7, the result obtained is 7.
- The square root of any negative numbers is not defined. Because the perfect square cannot be negative.
- If a number ends with 2, 3, 7 or 8 (in the unit digit), then the perfect square root does not exist.
- If a number ends with 1, 4, 5, 6 or 9 in the unit digit, then the number will have a square root.
Perfect squares
Below are the numbers which are perfect squares and then finding the square roots of such numbers is easy.
- 1^{2} = 1
- 2^{2} = 4
- 3^{2} = 9
- 4^{2} = 16
- 5^{2} = 25
- 6^{2} = 36
- 7^{2} = 49
- 8^{2} = 64
- 9^{2} = 81
- 10^{2} = 100
Hence, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 are the perfect squares here. Check square roots of some numbers here:
Square Root List
Here is the list of the square root of numbers from 1 to 50.
√n | Value | √n | Value | √n | Value |
√1 | 1 | √18 | 4.2426 | √35 | 5.9161 |
√2 | 1.4142 | √19 | 4.3589 | √36 | 6 |
√3 | 1.7321 | √20 | 4.4721 | √37 | 6.0828 |
√4 | 2 | √21 | 4.5826 | √38 | 6.1644 |
√5 | 2.2361 | √22 | 4.6904 | √39 | 6.2450 |
√6 | 2.4495 | √23 | 4.7958 | √40 | 6.3246 |
√7 | 2.6458 | √24 | 4.8990 | √41 | 6.4031 |
√8 | 2.8284 | √25 | 5 | √42 | 6.4807 |
√9 | 3 | √26 | 5.0990 | √43 | 6.5574 |
√10 | 3.1623 | √27 | 5.1962 | √44 | 6.6332 |
√11 | 3.3166 | √28 | 5.2915 | √45 | 6.7082 |
√12 | 3.4641 | √29 | 5.3852 | √46 | 6.7823 |
√13 | 3.6056 | √30 | 5.4772 | √47 | 6.8557 |
√14 | 3.7417 | √31 | 5.5678 | √48 | 6.9282 |
√15 | 3.8730 | √32 | 5.6569 | √49 | 7 |
√16 | 4 | √33 | 5.7446 | √50 | 7.0711 |
√17 | 4.1231 | √34 | 5.8310 |
How do Find Square Root?
To find the square root of any number, we need to figure out whether the given number is a perfect square or imperfect square. If the number is a perfect square, such as 4, 9, 16, etc., then we can factorise the number by prime factorisation method. If the number is imperfect square, such as 2, 3, 5, etc., then we have to use long division method to find the root.
Example: Square of 7 = 7 x 7 = 7^{2} = 49
The square root of 49, √49 = 7
Square root By Prime Factorisation
The square root of perfect square number is easy to calculate using the prime factorisation method. Let us solve some of the examples here:
Number | Prime Factorisation | Square Root |
16 | 2x2x2x2 | √16 = 2×2 = 4 |
144 | 2x2x2x2x3x3 | √144 = 2x2x3 = 12 |
169 | 13×13 | √169 = 13 |
256 | 256 = 2×2×2×2×2×2×2×2 | √256 = (2x2x2x2) = 16 |
576 | 576 = 2x2x2x2x2x2x3x3 | √576 = 2x2x2x3 = 24 |
How to Find Square Roots Without Calculator?
This is quite an interesting way to figure out the square root of a given number. The procedure completely based on the method called “guess and check”. Guess your answer, and verify. Repeat the procedure until you have the desired accurate result. We can also use the long division method to find the square root of a number.
Square root of Complex Numbers
To find the square root of complex numbers is a little complicated process. We can find the square root of a+ib using the below formula:
\(\sqrt{a+b i}=\pm(\sqrt{\frac{\sqrt{a^{2}+b^{2}+a}}{2}}+i \sqrt{\frac{\sqrt{a^{2}+b^{2}-a}}{2}})\)
where a+ib is a complex number.
How to Solve the Square Root Equation
To solve the square root equation, we need to follow the below steps: Isolate the square to one of the sides (L.H.S or R.H.S). Square both the sides of the given equation Now solve the rest equation. Let us understand the steps with examples.
Example: Solve √(4a+9) – 5 = 0
Solution: Given, √(4a+9) – 5 = 0
Isolate the square root term first. Then the equation becomes,
√(4a+9) = 5
Now on squaring both the sides, we get; 4a+9 = 5^{2}
4a + 9 = 25
4a = 16
a = 16/4
How to Square a Number?
To find the square of a number, we need to multiply the number by itself.
For example, 2 multiplied by 2 is equal to 4
Below is a 2 by 2 table that shows, total of four blocks.
1 | 2 |
3 | 4 |
Similarly,
square of 5 is: 5 multiplied by 5 = 5 x 5 = 52 = 25
Square of 9 = 92 = 9 x 9 = 81
Square of 15 = 152 = 15 x 15 = 225
Squares and Square Root
Let us see the value of squares and the square root of the numbers here.
Numbers | Squares | Square root |
0 | 0^{2} = 0 | √0 = 0 |
1 | 1^{2} = 1 | √1 = 1 |
2 | 2^{2} = 4 | √4 = 2 |
3 | 3^{2} = 9 | √9 = 3 |
4 | 4^{2} = 16 | √16 = 4 |
5 | 5^{2} = 25 | √25 = 5 |
6 | 6^{2} = 36 | √36 = 6 |
7 | 7^{2} = 49 | √49 = 7 |
8 | 8^{2} = 64 | √64 = 8 |
9 | 9^{2} = 81 | √81 = 9 |
10 | 10^{2} = 100 | √100 = 10 |
Applications
The square root formula is an important section of mathematics that deals with many practical applications of mathematics and it also has its applications in other fields such as computing. Some of the applications are:
- Quadratic equations
- Algebra
- Geometry
- Calculus
Solved Examples
Let us understand this concept with the help of an example:
Example 1: Solve √10 to 2 decimal places.
Solution:
Step 1: Select any two perfect square roots that you feel your number may fall in between. we know that 2^{2} = 4; 3^{2} = 9, 4^{2} = 16 and 5^{2} = 25
Now, choose 3 and 4 (as √10 lies between these two numbers)
Step 2: Divide given number by one of those selected square roots.
Divide 10 by 3.
=> 10/3 = 3.33 (round off answer at 2 places)
Step 3: Find the average of root and the result from the above step i.e.
(3 + 3.33)/2 = 3.1667
Verify: 3.1667 x 3.1667 = 10.0279 (Not required)
Repeat step 2 and step 3
Now 10/3.1667 = 3.1579
Average of 3.1667 and 3.1579.
(3.1667+3.1579)/2 = 3.1623
Verify: 3.1623 x 3.1623 = 10.0001 (more accurate)
Stop the process.
Example 2: Find the square roots of whole numbers which are perfect squares from 1 to 100.
Solution: The perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Square root | Result |
√1 | 1 |
√4 | 2 |
√9 | 3 |
√16 | 4 |
√25 | 5 |
√36 | 6 |
√49 | 7 |
√64 | 8 |
√81 | 9 |
√100 | 10 |
Example 3: What is:
- The square root of 2
- The square root of 3
- The square root of 4
- The square root of 5
Solution: Use square root list, we have
- value of root 2 i.e. √2 = 1.4142
- value of root 3 i.e. √3 = 1.7321
- value of root 4 i.e. √4 = 2
- value of root 5 i.e. √5 = 2.2361
Example 4: Is square Root of a Negative Number a whole number?
Solution: No, As per the square root definition, negative numbers shouldn’t have a square root. Because if we multiply two negative numbers, the result will always be a positive number. Square roots of negative numbers expressed as multiples of i (imaginary numbers).
Practice Problems
- Simplify √142
- Find the value of √12.
- Are √155, √121 and √139 perfect squares?
Frequently Asked Questions – FAQs
What is a square root?
How to find square root?
What are squares and square roots?
How to find the square root of perfect squares?
How to find the square root of imperfect squares?
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