Square Root

The positive number, when multiplied by itself, represents the square of the number. The square root of square of a positive number 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 x² = y. Here, ‘√’ is the radical symbol used to represent the root of numbers.

For example, the square of 3 is 9, 3² = 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.

Table of Contents:

• Definition
• Symbol
• Formula
• Property
• Perfect Squares
• Table
• Finding Square Root
  • Without Calculator
• Square Root of Complex Numbers
• Square Root Equation
• Squares and Square Root
• Applications
• Examples
• FAQs

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) = √(m²) = m

Note: The square root of a negative number represents a complex number.

Suppose √-n = i√n, where i is the imaginary number.

Square Root 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 itself. 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.

Square Root Formula

The formula to find the square root is:

Since, y.y = y² = a; where ‘a’ is the square of a number ‘y’.

Property of Square root

Also, the square root function of a number that helps to map a set of non-negative real numbers onto itself. It is usually represented as:

f(x) = √x

Consider a number ‘a’ acting as the square root of a number ‘y’ such that y² = a, where multiplying the number y with itself will lead to its square root (y.y).

Perfect squares

Below are the numbers which are perfect squares and the finding the square roots of such numbers is easy.

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• 6² = 36
• 7² = 49
• 8² = 64
• 9² = 81
• 10² = 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 Table

Here is the list of the square root of numbers from 1 to 50.

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 = 7² = 49
The square root of 49, √49 = 7

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. Mostly used to find the square roots of numbers that aren’t perfect squares. 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 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. √(4a+9) = 5 Now on squaring both the sides, we get; 4a+9 = 5²
4a + 9 = 25 4a = 16 a = 16/4 a = 4

Squares and Square Root

Let us see the value of squares and the square root of squares of the same numbers here.

Applications of Square Roots

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.
2² = 4; 3² = 9, 4² = 16 and 5² = 25
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. Answer!

Example 2: Find the square roots of whole numbers perfect squares from 1 to 100.
Solution: The perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Example 3: What is:

1. Square root of 2
2. Square root of 3
3. Square root of 4
4. Square root of 5

Solution: Use square root list, we have

1. value of root 2 i.e. √2 = 1.4142
2. value of root 3 i.e. √3 = 1.7321
3. value of root 4 i.e. √4 = 2
4. 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 result will always be a positive number.  Square roots of negative numbers expressed as multiples of i (imaginary numbers).

Practice Problems

1. Simplify √142
2. Find the value of √12.
3. Are √155, √121 and √139 perfect squares?

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