Square Root

Square root of a number is a value, which on multiplication 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 x² = 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, 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.

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.

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:

Since, y.y = y² = 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 an 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 radical 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.

• 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 List (1 to 50)

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 an imperfect square, such as 2, 3, 5, etc., then we have to use a long division method to find the root.
Example: Square of 7 = 7 x 7 = 7² = 49
The square root of 49, √49 = 7

Square root By Prime Factorisation

The square root of a perfect square number is easy to calculate using the prime factorisation method. Let us solve some of the examples here:

Click here to learn more about the prime factorization and methods.

How to Find Square Root by Division Method

Finding square roots for the imperfect numbers is a bit difficult but we can calculate using a long division method. This can be understood with the help of the example given below. Consider an example of finding the square root of 436.

Thus, the square root of 436 is 20.880 (rounding to 3 decimals).

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 is 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 Decimal

A decimal value will have a dot (.) such as 3.8, 5.2, 6.33, etc. For a whole number, we have understood how to derive the square root but let us see how to get the square root of a decimal.

Example: Find the square root of 0.09.
Let N² = 0.09
Taking root on both sides.
N = ±√0.09
As we know,
0.3 x 0.3 = (0.3)² = 0.09
Therefore,
N = ±√(0.3)²
N = ±(0.3)

Square root of Complex Numbers

To find the square root of complex numbers is a little complicated. 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

A square root equation is such an equation that has a variable in the radicand of the root. It is also called the radical equation.

To solve the radical equation, we need to follow the below steps:

• Isolate the square root 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 the radical equation √(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 = 5²
4a + 9 = 25

4a = 16

a = 16/4

a = 4

Condition 1: If the equation has more than one radical or square root.

If the radical equation has more than one root then repeat the above given steps for each square root.

Example: Solve √(2x−5) − √(x−1) = 1

Let us isolate one of the square root.

√(2x−5) = 1 + √(x−1)

Now squaring both the sides

2x – 5 = (1 + √(x−1))²

Apply the algebraic identity, (a+b)² = a² + b² + 2ab.

2x−5 = 1 + 2√(x−1) + (x−1)

2x−5 = 2√(x−1) + x

x−5 = 2√(x−1)

Now again isolate the square root.

√(x−1) = (x−5)/2

x−1 = ((x−5)/2)²

x−1 = (x2 − 10x + 25)/4

4x−4 = x² − 10x + 25

4x − 4 − x² + 10x − 25 = 0

−x² + 14x − 29 = 0

x² − 14x + 29 = 0

Using the quadratic formula, we can solve the above equation.

x = 2.53 and x = 11.47

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, a total of four blocks.

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.

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

Problems and Solutions

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 2² = 4; 3² = 9, 4² = 16 and 5² = 25
Now, choose 3 and 4 (as √10 lies between these two numbers)
Step 2: Divide the 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

Example 3: What is:

1. The square root of 2
2. The square root of 3
3. The square root of 4
4. The 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. This is 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).

Example 5: Solve the equation: √(x+2) = 4

Solution: Given,

√(x+2) = 4

Squaring both the sides, we get;

X+2 = √4

x+2 = ±4

x = ±4 – 2

Therefore,

x = 2 or x = -6

Practice Problems

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

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