Definition, Rules, Table, How to Find

Repeated subtraction method

The repeated subtraction method first subtracts the given number from the odd numbers until you get zero. The step in which zero is obtained is the square root of the number. It is a time-consuming and lengthy process. 
Find the square root of 36 using the repeated subtraction method

1. 36-1 = 35
2. 35-3 = 32
3. 32-5 = 27
4. 27-7 = 20
5. 20-9 = 11
6. 11-11 = 0

We got zero in the sixth step. Hence,  √36 = 6.

Square root calculation is used in finance, quadratic formula, standard deviation lengths and distance, and many more. With this blog, we will understand what is square root, the formula and symbol, and some solved examples.

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Before understanding what the square root of a number is, it is important to understand the meaning of root of a number.

The root of a number is an equal factor of the number. For example, here is how to find the root of 16.

First, we need to factor 16. The different ways to factor 16 are shown below.

16 = 1 × 16

16 = 2 × 8

16 = 4 × 4

The root of 16 is 4 because 4 is the equal factor for 16. We call 4 the

square root

of 16 and we write √16 = 4.

4 is called square root because we have to square 4 or raise 4 to a power of 2 to get 16.

What are the methods to find the square root of a number?

The numbers that end with 2,3, 7 0r 8 at the unit place are not perfect square numbers and are a little difficult to find. The numbers ending with 1,4,5,6 or 9 in the unit place are perfect square numbers and can be found easily using the prime factorization or the long division method. 
The method to find the square root is

• Prime factorization method
• Long division method
• Repeated subtraction method

Prime factorization method

Prime factorization is one of the easiest methods to find the square root of any number. Let us take n as the prime number, by grouping the similar numbers we get n², by multiplying the similar n^2. The value we get is the square root of the number. 
Let us take the number 144 as an example and solve them using the prime factorization method.
The prime factor of 144 = 12×12
By squaring them we get 12², since two is the only similar square we take the common one.
Here, the common square is 12.
Hence, the √144 = 12

Suppose you are asked to find the square root of a number. But for that, you need to know what are squares and square roots. The value of the number being multiplied by itself gives the original given number back, known as its square root. For example, let’s assume y is the square root of x, then y = x. This equation can also be written as y = √x. The square root of a number is represented by the symbol ‘√ ‘ Whenever we multiply a number by itself, the positive value we get is a square of the number. We get the original number back by finding out the square root of this positive value.

Long division method

• Step 1: Find the smallest integer that can divide the number. 
• Step 2: Keep following the long division using divisor and dividend. 
• Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number.

Find the square root of 2

• Step 1: Find the smallest integer that can divide the number. 1 is the perfect root number that is smaller and closest to 2.
• Step 2: Keep following the long division using divisor and dividend.

• Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number. Hence, the √2 = 1.4142

One way to estimate the square root of any number, especially numbers that are not perfect squares, is to find a whole number greater than the square root and another whole number less than the square root of that number that is not a perfect square. 

Take a close look at the figure below so you can learn the process. There are some important observations you need to make!

• In our example below, since 34 is not a perfect square, we can try to find an approximation for the square root.

• Perfect square numbers do not need estimation. You just need to know your multiplication table to find them. Many people know the perfect square numbers from 1 to 1000.

• Notice that 34 is much closer to 36 than it is to 25. Therefore, we chose a number very close to 6 and that number is 5.8.

The example in the figure above is a quick and easy way to estimate a square root when it is not possible to find perfect square roots. You will learn two different methods in this lesson.

Another estimation method

Estimate the square root of 45

Pick a number you think √45 is close to. For example, pick 7 since √49 = 7. We did not pick 6 since √49 is closer to the answer than √36.

Find the difference between the square of √49 and the square of √45

(√49)² — (√45)² = 49 — 45 = 4

Divide the number you found in step 2 by twice the number you picked in step 1

4 / 2(7) = 4 / 14 = 0.2857

Since 7 is an overestimation, subtract 0.2857 from 7 and this is your estimation.

7 — 0.2857 = 6.714

6.714 times 6.714 = 45.0777 and as you can see 45.0777 is quite close to 45

Estimate the square root of 39

Pick a number you think √39 is close to. For example, you can pick 6 since √36 = 6. 

Find the difference between the square of √39 and the square of √36

(√39)² — (√36)² = 39 — 36 = 3

Divide the number you found in step 2 by twice the number you picked in step 1

3 / 2(6) = 3 / 12 = 0.25

Since 6 is an underestimation, add 0.25 to 6 and this is your estimation.

6 + 0.25 = 6.25

6.25 times 6.25 =  39.0625 and as you can see 39.0625 is quite close to 39

Estimate the square root FAQs

The easiest way to calculate square root is to use a calculator. This is a must if the number you are taking a square root of is not a perfect square and in this cae, teachers will let you use a calculator in most cases. When taking exams, it is not likely they will force you to use paper and pencil only if you are not dealing with perfect square roots.
If you are taking the square of numbers that are not perfect squares, you will end up with irrational numbers with a large number of digits after the decimal point. You can just round the answer to the tenths place or hundredths place.

√22 lies between 4 and 5. Since √22 is closer to 5, you could choose 4.7 as an estimate.

√23 lies between 4 and 5. Since √23 is closer to 5, you could choose 4.8 as an estimate.

We calculate the area of a square as a product of side i.e , but if we have to calculate the side of a square we need to take the square root of the area. Thus we can say that the square root is the inverse operation of squaring a number. Square roots are used in solving algebraic equations and other complex Mathematical and Scientific calculations.Square root of a number has two values, positive and negative. Now let us study root finder methods to find the square root of imperfect squares.

What are Square Roots?

Finding Square Roots:

• Prime factorization method

• Repeated subtraction method

• Long division method

• Number line method

But, if the number is not a perfect square prime factorization method and the repeated subtraction method will not work, we have to use other methods for finding the square roots. Let us study how to calculate imperfect square roots.

Finding Square Roots of Imperfect Squares

How to find the square root of imperfect numbers by the average method: 

Find out the two perfect square numbers which are very close to the given number on either side. For example the number ‘10’, the immediate perfect square lesser than 10 is ‘9’ and the immediate perfect square greater than 10 is ‘16’.

Step 2 : Note down the square roots of the perfect squares, here the square root of ‘9’ is ‘3’ and the square root of ‘16’ is ‘4’.

Step 3 : Square root of a given number lies between the square roots of numbers determined in step 2.Square root of ‘10’ is any number between 3 and 4.

Divide the number whose square root is determined by any of the numbers obtained in Step 2.‘10’ can be divided either by ‘3’ or ‘4’.

Let us divide ‘10’ by ‘3’

Find the average of the quotient and divisor in Step 4.

The average of 3 and 3.33 is

Now divide 10 by step 5 answer

Now, average 3.1579 and 3.1667 by adding them together and dividing the sum  by two you get 3.1623. Check your work by multiplying your answer by itself. If 3.1623 is multiplied by 3.1623 we get 10.001.

How to find the square root of imperfect numbers by long division method

Let us find the square root of 104976 by using a long division method. 

Step 1 :

Separate the numbers by taking commas from right to left in a group of two digits.

Step 2 :

Now we have to multiply a number by itself such that the product is less than or equal to 10

here, 3 x 3  =  9  will meet the condition

Now, 9 is subtracted from 10 and we got the remainder 1.

Step 3 :

Now, we have to bring down 49 so the remainder becomes 149 and add 3 to quotient 3 we get 6.

Step 4 :

Then, we have to find a product such that it is less than 149 and the same number will be written in the divisor column in quotient.

Step 5 :

The condition said in step 4 will be met by «2». 62 multiplied by 2 we get 124. We get remainder 25.

Step 6 :

Now, we have to bring down 76 and add 2 to 62. We get the remainder of 2576 and divisor 64.

Step 7 :

Now multiply 644 by 4 we get 2576. So the remainder becomes zero.

Step 8 :

Finally, we got the square root of 104976  is 324.

Below figure represents the long division method. From these steps it is clear how to calculate imperfect square roots.

Properties of Square Root

• Only a perfect square number has a perfect square root.

• The square root of an even perfect square is even.

• Because a perfect square cannot be negative, the square root of a negative number cannot be defined.

• A number cannot have a square root if it finishes with an odd number of zeros. A square root can only be calculated with an even number of zeros.

• In a set of real numbers, negative values have no square roots.

• A square root is found in numbers that conclude with (containing a unit’s digit) 1, 4, 5, 6, or 9.

What is square root? A number multiplied by itself is called the square of that number. So $x \times x=x^2$ is the square of $x.$ Here the number $x$ is called the square root of $x^2.$

Definition of Square Root

Thus, if $y=x^2$ then we say $x$ is the square root of $y.$

Symbol of Square Root

Square Root Examples

Few examples of square roots:

Rules of Square Root

We list all the properties of square roots.

• $x^{1/2}=\sqrt{x}$
• $\sqrt{x^2}=x$
• $a\sqrt{x}+b\sqrt{x}=(a+b)\sqrt{x}$
• $a\sqrt{x}-b\sqrt{x}=(a-b)\sqrt{x}$
• $\sqrt{x \times y}=\sqrt{x} \times \sqrt{y}$
• $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$   if $y \neq 0$

Square Root Formula

Square Root Table

Square toot table: (perfect square)

Square Root as a Function

Note that $f$ is a one-to-one function but not onto.

Methods of Finding Square Root

• Square root by prime factorization
• Square root by repeated subtraction method
• Square root by estimation method
• Square root by long division method

Simplifying Square Roots 

Now, we will learn how to simplify square roots. Let us simplify the square root of $27$.

Firstly, we will factorize $27.$ Note that

$27=3 \times 3 \times 3$

Taking square root on both sides, we get that

Addition or Subtraction of Square Roots 

First, we express the square roots into their simplified forms as above.

Firstly, we will simplify both square roots.

Multiplication or Division of Square Roots 

First, we express the square roots into their simplified forms as above.

Square Root of a Decimal

At first, we have to express the decimal number as a fraction.

Now we will apply the method of finding the square root of real numbers to compute √x and √y.  And we put their values in the fraction obtained in step 2.

At last, we will simplify the fraction obtained in step 3. The resultant value will be the square root of the given decimal number.

Let’s understand the above process by an example.

Taking square root we get

So the square root of $0.16$ is $0.4$

Square Root of a Negative Number

We know that the square of any number is always positive. So by definition, the square root of a negative number does not exist in the set of real numbers. But they do exist in the world of complex numbers.

Square Root of a Complex Number

The computation of the square root of a complex number is not as simple as the method of finding the square root of a real number. We know that the general form of a complex number is $a+ib,$ where both $a$ and $b$ are real numbers. The formula for finding the square root of $a+ib$ is given below:

Using the above formula, we can find the square root of $i.$ Note that $a=0$ and $b=1$ as $i=0+i \cdot 1.$ So we have

In a similar way, one can find the square root of $-i$ also.

Square Root of a 2×2 matrix

Square root of a matrix: Let $A$ and $B$ be two matrices such that $A^2=B.$ In this case, we say that the matrix $A$ is the square root of $B.$ Note that there may have many square roots of $B,$ for example, $I_2,$ the 2×2 identity matrix, has infinitely many square roots. So it is natural to ask the question: is the square root of a matrix well-defined? The answer is yes. We will discuss it now.

Let $B$ be a positive-definite matrix. We know that there is only one positive-definite matrix $A$ such that $A^2=B.$ So we define

The square root of $B$ is well-defined. We note that among the infinite number of square roots of $I_2,$ there is exactly one positive-definite which we define to be the square root of $I_2.$

Generalization of Square Roots

Let $f(x)$ be a polynomial. If a number $c$ satisfies the equation $f(c)=0,$ then $c$ is called a root of $f(x).$ This type of roots are known as polynomial roots.

Square Root Applications

The computation of square roots has many applications in several branches of mathematics; such as

• Equation solving
• Polynomial
• Ring theory, field theory
• Numerical analysis
• Geometry (for example, to find the side of a square if the area is known)

Solved Problems of Square Roots

Find the square root of $125.$

Note that $125=5 \times 5 \times 5$

How to Find Square Root of Numbers

Are you taking extra time than it actually requires to solve complex square root equations? Well, now with the help of square root tricks you can find the square of numbers very easily and with much less time. Square roots are very important from an examination point of view as it is needed in almost every topic. Tips and tricks always help us to solve mathematical problems easily and swiftly. Therefore, we have here for you a few helpful tips with which you can find the square root of a given number without any kind of help, especially that of a calculator. Without knowing these tips and tricks, it can be time-consuming to find a simple square root of a number. 

To begin with, finding the square root of a number, what we have to know first is if the number is a perfect square or not. And we all know that there are two conditions to find if a number is a perfect square or not. First, a number will be a perfect square if it ends with 1, 4, 5, 6, and 9. Second, a number will never be a perfect square if it ends with 2, 3, 7, and 8. These two tips are the most basic tips to find square roots but they are not enough. 

(Image will be uploaded soon)

What is a Square Root?

The square root of a number is a value that we get when it is multiplied by itself and produces the original number. For example, when 5 is multiplied by itself we get 25. Thus we can say that 5 is a square root value of 25. In the same way, 4 is the square root value of 16, 6 is the square root value of 36, and 7 is the square root value of 49. 

Now, just like a square is a representation of the area of a square that is equal to the side x side, the square root is the representation of the length of the side of a square. 

The square root of a number can be  rational  or irrational. If the square root of a number is an integer, it is a perfect square. 

How to Find the Square Root of a Number?

Finding the square root of numbers such as 4, 9, 16, 25, etc. is quite easy, and I’ll tell you why. It is because we all know that from the multiplication table of 1 to 10, the number which is multiplied by itself gives the squares in a two-digit form. But what if a number is in three-digits or four-digits? Well, then it is considered difficult to find the root of these numbers. And that is because we fail to remember the table for higher numbers. So why not know the trick behind to determine the root of larger numbers?

Tricks to Calculate Square Root 

You can use the trick to estimate the square root of the perfect square number. To find the square root without long division, you need to know the square root of the first 10 numbers. The square roots of the first 10 numbers are- 

Square Root Trick for 4 Digit Numbers

Pair the numbers from the right hand side.  

Compare the unit digits of  the numbers in the figure to determine the possible values ​​for the square root of the unit digits. 

Let’s take a look at the first pair of numbers. Let’s say «n».  

Determine the two squares where this number is between √a < n < √b, then a < n < b. Therefore, the tens digit of square roots of the required is «a».  

As shown in the square table, there are only two numbers and the square does not repeat which is 5 or 10. Confirm that the unit number obtained in step 2 is one of them.  

 Multiply a and b. 

Select b if ab ≤ n, otherwise select a. 

 Let us consider that we need to find the square root of a large number of 4489.

• Here, in this number, the unit digit is 9 that means it can be a unit digit of its square root number that is 3 or 7 that is because 32 is 9 & 72 is 49.

• Now if we consider the first two digits, that is 44, it comes between the squares of 6 and 7 because 62 < 44 < 72.

• We can expect that in the ten’s digit of the square root of 4489,the lowest amidst the two numbers is 6 and we want to find the unit digit of the square root of the number 4489.

• Now, we would want to find if 63 or 67 is the square root of 4489.

• Considering the ten’s digit is 6 and the next number is 7, we have to multiply both the numbers like 6 x 7 = 42 and because 42 is less than 44.

• Square root of 4489 has to be the bigger number between 63 and 67 i.e. 67.

We can have a look at one more example, the square root of 7056.

Given below is the step by step method:

• Now, in this number, the unit digit is 6. All the numbers which have the unit digit as 6 on their square roots. are 4 and 6.

• Now consider that the first two digits i.e., 70 come between the squares of 8 and 9 because of 82 < 70 < 92.

• Assume that in the ten’s digit of the square root of the 7056, the lowest amidst the two numbers is 8.

• So, we have to find the unit digit of the square root of the number 7056. And for that, we have to find between 84 and 86 which one is the square root of 7056.

• Since the ten’s digit is 8 and the proceeding number is 9, we have to perform multiplication of both the numbers like 8 x 9 = 72 and because 72 is greater than 70.

• The square root of 7056 needs to be a lesser number between 84 and 86 that is 84.

There are various square roots tricks pdfs which contain more tricks just like this method that you can find on the web. Try to find the square roots of large numbers using these tricks, and you will be able to solve an equation within no time.

Given below is a table of square roots from numbers 1 to 50. This table will help you to solve the problems based on them very easily.  

Square Root Table From 1 to 50

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The square root of the numbers from 1 to 20

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Solved examples

Q1: Solve the equation √2a+9 = 5
A1: √2a+9 = 5
Let us take square on both sides
2a+9 = 5²
2a+9 = 25
2a = 25-9
2a = 16
a = 16/2
a=8

Q2: Square root of 125 using the prime factorization method
A2: The prime factors of 125 = 5×5×5
√125 = 5×5
= 5 × 2.236
= 11.8

Q3: Square root of 12 using the long division method

A 3: √12 = 3.464

Q4: Find the square root of pi

A 4: √𝛑 = 1.77

Q5: Find the square root of 64 using the repeated subtraction method

1. 64-1 = 63
2. 63-3 = 60
3. 60 – 5 = 55
4. 55-7 = 48
5. 48 – 9 = 39
6. 39-11 = 28
7. 28-13 = 15
8. 15-15 = 0
9. √64 = 8

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Can the square root of a number be a real number?

For the numbers above, the square root was equal to an integer.

it is not always possible to get the square root as an integer.

Sometimes, you may get a real number when finding the square root.

For example, use the square root calculator below to find the square root of 5.

The result includes lots of numbers after the decimal point.

Ready for big time challenge? Just like long division, learn how to compute the square root without a calculator for any number that is not a perfect square.I promise you will not sweat too much!

Teachers! Do you want a ready made square roots table that students can quickly refer to as they solve their basic math problems? Get the square roots table.

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Frequently asked question

What is a square root?
The square root of a value is a number when multiplied by itself gives the original number or the square root is the inverse of squaring the numbers. Which tells that both concepts are dependable on each other

Give an example of a perfect square number.
The numbers that are perfect squares are 4,16 and 25 are some of the examples for perfect square numbers.

What are irrational numbers?
Any number that is not expressed as a fraction is called an irrational number.

What is the square root of -1?
We know the definition of square root, thus multiplying or squaring these values we get 1 and not a negative result.
-1 = (-1) (-1) = 1. The square root of -1 is i

What is a radicand?
The square root symbol is denoted as √ called a radicand.

What is the formula for square root?
The formula is simple x = y or x = y²

What is the square root of 100?
The square root of 100 is 10

Can the square root of a number be negative?

Yes, absolutely! If you multiply a positive number by itself, you get a positive product. If you multiply a negative number by itself, you also get a positive product.

For example, since -8 × -8 = 64, -8 is also a square root of 64. However, unless otherwise stated, the square root sign (  refers to the positive root of a number, also called principal square root.

What is a Square Root?

The square root of a number is another number that when multiplied by itself equals this number. For example, the square root of 25 is 5 because 5 × 5 = 25.

Instead of writing ‘find the square root of’ in front of each number, it is quicker to use the square root sign.

The mathematical symbol for finding the square root is √. It is written immediately before the number that is to be square rooted. For example, √36 means to find the square root of 36. It means to find the number that when multiplied by itself equals 36. √36 = 6 because 6 × 6 = 36.

Here is a list of square roots:

Here is a downloadable, printable poster showing the first 12 whole number square roots.

Square roots are called square roots because any square has a side length that is equal to the square root of the area. The relationship between a number and its square root can be shown using a square. The word root simply means a solution to an equation.

For example, here is a square of area 25.

The area of a square is the length of one of the sides multiplied by itself.

The length of each side of the square is 5. The area of the square is 5 × 5 = 25.

When a number is multiplied by itself we say that the number has been squared.

We say that 5 squared is 25.

Finding the square root is the opposite of squaring a number. They are inverse functions. Squaring means to multiply a number by itself. Finding the square root means to find the number that can be multiplied by itself to give that number.

Since we know that 5 × 5 = 25, we know that 5 is the square root of 25.

The square root of a given number can be thought of as the side length of a square which has an area equal to that given number.

To find the square root of a given number, find the number that can be multiplied by itself to equal the given number. Knowing the times tables is helpful for finding the square roots more quickly.

A perfect square root is a square root that is a whole number.

Here is a list of perfect square roots:

• √1 = 1.
• √4 = 2.
• √9 = 3.
• √16 = 4.
• √25 = 5.
• √36 = 6.
• √49 = 7.
• √64 = 8.
• √81 = 9.
• √100 = 10.
• √121 = 11.
• √144 = 12.

Perfect square roots are the square roots of square numbers. They are always whole numbers.

When introducing the idea of square roots, it is best to start with finding the square roots of perfect squares. Start by teaching the square numbers by multiplying a number by itself. Use these square number facts to then work out the square roots.

You can find the square root of all numbers except for negative numbers. Negative numbers do not have a square root. This is because when a number is multiplied by itself, the answer is always positive. Even a negative number multiplied by itself makes a positive answer because two negatives multiplied together cancel out.

The square root of 0 is 0 because 0 × 0 = 0.

To find the square roots of non perfect squares, the easiest and most common method is to use a calculator. Simply press the √ key, type the number and press the = key.

There is a square root calculator on this page which can be used to find the square roots of non perfect squares.

The square roots of non perfect squares will be between the square roots of the perfect squares either side of the number.

For example, √9 = 3 and √4 = 2 so the square root of 5 must be less than 3 but larger than 2. The square root of 5 is actually 2.2360679775, which is less than 3 but larger than 2.

Square root

The square root of a value is a number when multiplied by itself gives the original number or the square root is the inverse of squaring the numbers. Which tells that both concepts are dependable on each other.

The square root of a number in the radical form is denoted as √ when they are in the exponential form they are denoted as the (number)^1/2 or the (number) ^0.5.

Let us take p and q. The square root of p = √q or p = q² the number that is positive when positive or multiplied gives the exact number.
That is if we take the number 9 the square root of 9 is found by multiplying the number twice.
9 = 3² or √9 = 3, when the number three is multiplied it gives the number 9.

Formula to Find Square Root of A Number

The square of 4 is 16 or 4 = 16 and √16 = 4. It is easy to find the square root of 16 or such numbers because they are perfect squares. But for the numbers like 10, 7, 20, 24, and many more, it becomes a bit difficult and calculative. There is no such formula to find the square root of a number. But there are various methods and tricks by which calculation of square roots can be easy. In different sections, we will learn about them.

Representation of Square Roots

The symbol used to represent the square root is known as the radical symbol. ‘√’ is the square root symbol. If x is the original number, then √x is the square root of the number. Here x is also known as the radicand. Radical and square root both represent the same thing. The square root formula can be given by:

X = √Y, where X is the square root of Y. On the other hand, X = Y. Therefore, where Y is the square of X. 

Properties of Squares And Square Roots

In Math, the square root function takes a positive value as input and gives the input value’s square root as the output. Hence, it is said to be a one to one function. For example, 

F(a) = √a.

Let us consider if x = 25, then the value returned will be 5. Below we have some of the properties of squares and square roots.

• A number can have a square root if it ends with an even number of zeros.

• A number can be a perfect square. Then, there will be a perfect square root also.

• Multiplication of two square root values is possible. For example, √2 × √3 = √6.

• If you find square root of a negative number, you get a complex number as a result. Perfect squares are never negative. 

• We get a radical number by multiplying two same square roots. If you multiply √7 × √7, you get 7 back, which is a nonsquare root number.

• Numbers ending with 1, 4, 5, and 6 will have a square root.

• Perfect square root does not exist for numbers ending with 2, 3, 7, and 8.

Easy Way to Find Square Root of A Number

We need to find the square root of a number, whether it’s a perfect or imperfect square. You already know if it’s an ideal square, then we can use the prime factorization method to factorize and find the square root easily. For other numbers, we have the division method.

Prime Factorization Method

It’s one of the easiest ways to find the square root of a number. For example, let’s take the number 256. Now break 256 into its prime factors. Therefore 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Now we need to pair in doubles of similar numbers and consider them once only, (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2). Hence, √256 = (2 × 2 × 2 × 2) = 16. 

Long Division Method

It is the square root formula used for imperfect squares or large numbers. Let’s consider number 3. It is not a perfect square number. Taking this example, we have illustrated the long division method below.

Writing 3 with a decimal point and groups of two zeros for the decimal. Putting all of this under the long square root symbol (which looks like division), and we start dividing.

1√3.00 00 00 00 00

Putting the largest number whose square is less than or equal to 3 and above the 3. Square it and subtract it from 3, which in this case is 1. 

Bringing down a pair of zeros and multiplying the answer so far by two. The next digit in the answer will be put at the end of the doubled answer and multiply that by the new digit. This product is subtracted, and two more zeros are brought down. We have to continue this process for as many digits of accuracy as we want.

The square root of 3 results in 1.732.

Now let us consider the number 24.

1. Give Some Applications of the Square Root Formula.

The square root formula is used in geometry, calculus, algebra, quadratic equations, computing, and other mathematical applications.

2. Give the Square Roots of 5, 7, 9, and 11.

The square roots of 5, 7, 9, and 11 are 2.236, 2.646, 3.0, and 3.317, respectively.

The square root of -1

Understanding the square root of minus values is simple. There is no real square root for the negative integers which are often referred to as imaginary roots. Why? Because what happens when you multiply two negative values is you get a positive number and not a negative one. Let’s say we need to find the square root of -1. We know the definition of square root, thus multiplying or squaring these values we get 1 and not a negative result.
-1 = (-1) (-1) = 1.
Therefore they are referred to as imaginary roots and are denoted as I.
Find the square root of -2
√-2 = 1.4142i

Rational and irrational numbers

Rational numbers are expressed as a fraction of the integer. That is they are simple fractions and p/q ≠0. 
Let us take the number 121 as an example:
The square root of 121 = 11, when we divide the number 121/11 = 11 and not zero or a complicated fraction.
An irrational number leads to complicated fractions and is often difficult to solve. Any number that is not expressed as a fraction is called an irrational number.
The square root of 5 = 7.071 which cannot be further converted into a fraction and is also not equal to 0.

Symbol of the square root

The square root symbol is denoted in a radical way. Let us say we have to express x in the square root then it is denoted as x. The symbol in radical form is also called radicant. Here x = any number.
The square root of 4 is expressed as √4 

The formula for finding the square root

The formula is simple x = √y or x = y²

Perfect square and not a perfect square

A perfect square number is one whose square roots give a whole number. The square root of 4 is 2, the square root of 400 is 20, etc.,
A not-perfect square number is one whose roots are not whole numbers but rather decimals.
The square root of 5 = 7.071

Other examples showing how to find the square root of a number.

Find the square root of 4.You can factor 4 in two different ways.

4 = 1 × 4

4 = 2 × 2

The equal factor is 2, so 2 is the square root of 4 and we write √4 = 2.

Find the square root of 64.

64 = 1 × 64

64 = 2 × 32

64 = 4 × 16

68 = 8 × 8

The equal factor is 8, so the square of 64 is 8 and we write √64 = 8.

More examples showing how to quickly estimate the square root

We will illustrate the process with two more examples.

Estimate the square root of 17

We will find a whole number bigger than the square root of 17 and a whole number smaller than the square root of 17. 

      <    √17    <

Study carefully the procedure!

First, let us find the number that is bigger than  √17

√17     <     √18

√17     <     √19

√17     <     √20

√17     <     √21

√17     <     √22

√17     <     √23

√17     <     √24

√17     <     √25

Notice that the square roots of 18, 19, 20, 21, 22, 23, 24 are all bigger than the square root of 17.

However, only √25 will give a whole number, so this is the one we will choose.

√25 = 5 since 5 × 5 = 25

Second, let us find the number that is smaller than √17

√16   <   √17

Since  √16  is a whole number, this is the one we will choose.

√16   = 4 since 4 × 4 = 16

We get 4  < √17  < 5

The square root of 17 is between 4 and 5. We could estimate the square of 17 to be 4.1 for example.

Estimate the square root of 102

We will find a whole number bigger than the square root of 102 and a whole number smaller than the square root of 102. 

√101   <   √102

√100   <   √102

We will use square root of 100 since  √100   = 10

To find the number bigger than the square of 102, we will use a different strategy. It take too long to write down the square root of all these numbers. Some good observation will help us to solve the problem quickly.

Notice that the square root of any number between 103 and 120 is not a whole number.

However, square root of 121 is a whole number since 11 times 11 = 121.

Therefore, the square of 121 will give us the whole number that we need that is bigger than square of 102.

We get 10   < √102   < 11

The square root of 102 is between 10 and 11. We could estimate the square root of 102 to be 10.2.

Notice again that in our estimation, we chose a number close to 10 since 102 is much closer 100 than it is 121.