Definition, Rules, Table, How to Find

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

Few examples of square roots:

Rules of Square Root

We list all the properties of square roots.

Square Root Formula

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

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

Firstly, we will factorize $27.$ Note that

$27=3 imes 3 imes 3$

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

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

Solved Problems of Square Roots

Find the square root of $125.$

Note that $125=5 imes 5 imes 5$

How to Find Square Root of Numbers

The «nth Root» used n times in a multiplication gives the original value

Instead of talking about the «4th», «16th», etc, we can just say the «».

The nth Root

So it is the general way of talking about roots
(so it could be 2nd, or 9th, or 324th, or whatever)

The nth Root Symbol

This is the special symbol that means «nth root»,
it is the «radical» symbol (used for square roots) with a little n to mean nth root.

Using it

We could use the nth root in a question like this:

Question: What is «n» in this equation?

Answer: I just happen to know that 625 = 54 , so the 4th root of 625 must be 5:

Or we could use «n» because we want to say general things:

Example: When n is odd then  

= a

  (we talk about this later).

Now we know what an nth root is, let us look at some properties:

Multiplication and Division

We can «pull apart» multiplications under the root sign like this:

=

×

(Note: if n is even then a and b must both be ≥ 0)

This can help us simplify equations in algebra, and also make some calculations easier:

Example

=
=

×
=
4

So the cube root of 128 simplifies to 4 times the cube root of 2.

It also works for division:

=

=
1/4

So the cube root of 1/64 simplifies to just one quarter.

Addition and Subtraction

But we cannot do that kind of thing for additions or subtractions!

Example: Pythagoras’ Theorem says

So we calculate c like this:

c = a2 + b2

Which is not the same as c = a + b , right?

It is an easy trap to fall into, so beware.

It also means that, unfortunately, additions and subtractions can be hard to deal with when under a root sign.

Exponents vs Roots

An exponent on one side of «=» can be turned into a root on the other side of «=»:

If  an = b  then  a =

Note: when n is even then b must be ≥ 0

54 = 625  so  5 =

Nth Root of a-to-the-nth-Power

Did you see that −3 became +3 ?

Here it is in a little table:

Nth Root of a-to-the-mth-Power

What happens when the exponent and root are different values (m and n)?

Well, we are allowed to change the order like this:

So this:    nth root of (a to the power m)
becomes  (nth root of a) to the power m

=

= 32
= 9

Easier than squaring 27 then taking a cube root, right?

The new exponent is the fraction which may be easier to solve.

=
= 42
= 16

This works because the nth root is the same as an exponent of (1/n)

You might like to read about Fractional Exponents to find out why!

318, 2055, 319, 317, 1087, 2056, 1088, 2057, 3159, 3160

The number system is the system defined for the different numbers and the way they could be arranged. There are many types of number systems but mostly 4 types are well known. They are Binary number systems, Decimal number systems, Octal number systems, and Hexadecimal number systems. The decimal number system is mostly used in mathematics, it involves numbers from 0-9. There are multiple operations done on the numbers, for example, finding squares and square roots of numbers, lets learn in detail about the square roots of the numbers,

The square root of a number is a value which when multiplied with itself gives the original number. For example, the square root of 9 is 3, when 3 is multiplied by itself, the original number obtained is 9. The symbol that denotes square root in mathematics is √.

This symbol (√) is called radical and the number inside the radical symbol is known as radicand. The number or the value present inside the root symbol might be a perfect square or an imperfect square. For example- 4 is a perfect square whereas 3 is an imperfect square. So, based on the nature of value inside the root, the final answer or the square root may be a natural number of a decimal number.Now let’s find out how to calculate the square root of different numbers.

Square roots without a calculator

As above defined, the square root of a number is the value which when multiplied with itself will provide the original number only. There are three ways to find square root without a calculator

This is a long but simple method to find the square root of any number. Prime factorization involves finding factors of that number and then pairing the common numbers in a pair of two. finally, taking the square roots of the prime factors. Let’s see an example of this,

Question: find the square root of 484

484=2 × 2 × 11 × 11

So, √484= √(2 × 2 × 11 × 11) = 2 × 11 =22

Guess & Check method

This method is used to give the approximate value of any number. The guess method saves time as it gives an approximate range of values between which the root exists. it is more efficient when the number inside the root is an imperfect number. Let’s see an example of this,

Question: find the square root of 20.

Start guess and check method by noting that since √16 = 4 and √25 = 5, then √20 must be between 4 and 5. As second step, in order to reach nearer to the actual answer, lets take a number between 4 and 5. lets assume it to be 4.5. Lets do square of 4.5 which comes out to be 20.25, which is greater than 20, therefore the root must be smaller than 4.5, lets choose 4.4, square of 4.4 is 19.36. thus, the most approx and accurate root of 20 is 4.4

Long division method

It is a very easy way to get the square root of imperfect squares. The long division method is mostly preferred over the other methods since it provides an accurate answer. Let’s understand this algorithm using an example,

Question: find the square root of 627

Step 1 Group the numbers in pairs from right to left ,leaving one or two digit in left (here its 6).

Step 2 Think of a number whose square is less than the first number (6), its 2, So,write it like this –

Step 3 Is to square the number 2 and write the result beneath 6 and then subtract as shown below,

Step 4  Multiply the quotient by 3 and and write it down in parenthesis with an empty line next to it as shown below,

Step 5 Now find out the number which when multiplied by forty something would be lesser than 225. Lets guess 5. then            45×5=225, which is less than 227, So write it as shown below-

Step 6 Then repeating step 4, multiply the quotient with 2 write it down in parenthesis with an empty line next to it as shown below,

Step 7 Repeating step 5, find out the number which when multiplied by five hundred something would be lesser than 2000. Lets guess 5, then 505×5=2525, which is bigger than 2000, lets guess 4, then 504×3=1512. So write it as shown below,

The square root of 627 with two decimal place is 25.03, which is accurate.

Sample Problems

Question 1: Find the square root of 144

144=2 × 2 × 2 × 2 × 3 × 3

So √144= √(2 × 2 × 2 × 2 × 3 × 3) = 2 × 2 × 3 =12

Question 2: Find the square root of 169

169=13 × 13

So √144= √(13 × 13) = 13

Question 3: Find the square root of 6 by the Guess and check method.

Start guess and check the method by noting that since √9 =3 and √4 = 2, then √6 must be between 2 and 3. As the second step, in order to reach nearer to the actual answer, let’s take a number between 2 and 3. Let’s assume it to be 2.5. Let’s do a square of 2.5 which comes out to be 6.25, which is greater than 6. therefore the root must be smaller than 2.5. Let’s choose 2.4, square of 2.4 is 15.76. Thus, the most approx and accurate root of 6 is 2.4

The square root of any numerical value is a value that on self multiplication results in the original number. ’√’ is the radical symbol used to depict the root of any number. By square root, we mean a power 1/2 of that number. For instance, let us suppose that x is the square root of any integer y, this implies that x=√y. On multiplying the eq, we also obtain x2 = y.

The square root of the square of a positive number gives the original number.

To understand the concept, we know, the square of 4 is 16, and the square root of 16, √16 = 4. Now, as we can see, 16 is a perfect square figure. This makes it easy to compute the square root of such numbers. However, to compute the square root of an imperfect square like 3, 5, 7, etc, computing root is a difficult process.

A square root function is a one-to-one function that uses as input a positive number and returns the square root of the given input number.

f(x) = √x

Properties of Square Roots:

How to calculate a square root?

Perfect square numbers are integers that are positive in nature and can be easily expressed in the form of the multiplication of a number by itself. Perfect square numbers are depicted as the value of power 2 of any integer. Computation of square root of perfect square numbers is relatively easier. There are primarily four methods used to find the square root of numbers:

The above three methods can be used in the computation of the square root of perfect square numbers. The last method, however, can be used for both types of numbers.

Repeated Subtraction Method of Square Roots

Step 1: Subtract consecutive odd numbers from the number for which we are finding the square root.

Step 2: Repeat step 1 until a value of 0 is attained.

Step 3: The number of times step 1 is repeated is the required square root of the given number.

Note: This method can be used only for perfect squares.

16 – 1 = 15

15 – 3 =12

12 – 5 = 7

7- 7 = 0

The process is repeated 4 times. Thus,√16 = 4.

Square Root by Prime Factorization Method

Step 1: Divide the specified number into its prime factors.

Step 2: A pair of similar factors is formed in a way such that both the factors in each of the formed pairs are equal.

Step 3: Take one factor from each of the pairs.

Step 4: The product of the factors is obtained by taking one factor from each pair.

Step 5: This obtained product is the square root of the given number.

64 = 22 × 22 × 22

64 = (2 × 2 × 2)2

64 = (8)2

√64 = 8

Square Root by Estimation Method

The estimation method is used for approximating the square root of a given number. It approximates the square root of a number to a reasonable guess of the actual value. Calculations are easier in this method. However, it is a really long and time taking process.

Step 1: Find the nearest perfect square occurring both before and after to the given number.

Step 2: Find the next closest integers and round them off each time to come towards the closest answer.

9 and 16 are the perfect square numbers before and after nearest to 15. Now, we know,

√16 = 4 and √9 = 3. This implies that the square root of the number 15 occurs between 3 and 4. Now, the process involves the evaluation of whether the square root of the number 15 is closer to 3 or 4.

The first case is taking 3.5 and 4. Square of 3.5 = 12.25 and the square root of 4 = 16. Therefore, the square root of integer 15 lies between 3.5 and 4 and is closer to 4.

Further, we find the squares of 3.8 and 3.9, which are equivalent to 3.82 = 14.44 and 3.92 = 15.21 respectively. This implies that √15 lies between 3.8 and 3.9. On further evaluation, we obtain that √15 = 3.872.

Square Root by Long Division Method

The Long Division method for the computation of the square root of numbers involves the division of large numbers into steps or parts, thus breaking the problem into a sequence of easier steps.

Step 1: A bar is placed over every pair of digits of the number beginning with the unit’s place.

Step 2: The left-most number is then divided by the largest number such that the square is less than or equal to the number in the left-most pair.

Step 3: Now, the number under the next bar to the right of the remainder is brought down. The last digit of the quotient obtained is added to the divisor. Now, the next step is to find a number to the right of the obtained sum, such that it together with the result of the sum, forms a new divisor for the new dividend.

Step 4: The obtained number in the quotient is equivalent to the number as selected in the divisor.

Step 5: The same process is repeated using a decimal point and adding zeros in pairs to the remainder.

Step 6: The quotient forms the square root of the number.

Sample Questions

Question 1. Compute the square root of 144 by Prime Factorization Method?

144 = 22 × 22 × 32

144 = (2 × 2 × 3)2

144 = (12)2

√144 = 12

Question 2. What is the way to simplify square root?

√xy = √(x × y), where, x and y are positive integers.

For instance, √12 =

Question 3. Solve: √(x + 2) = 4

On squaring both the sides, we obtain;

x + 2 = √4

x + 2 = ±4

x = ±4 – 2

Therefore, we have,

x = 2 or x = -6

Question 4. Can the square root of a negative number be a whole number? Explain.

We know, the negative numbers cannot have a square root. The reason behind this is that if two negative numbers are multiplied together, the result obtained will always be a positive number. Therefore, the square root of a negative number will be in the form of complex number.

Question 5. Compute the square root of 25 by the method of repeated subtraction?

Going by the above stated steps, we have,

25 – 1 = 24

24 – 3 = 21

21 – 5 = 16

16 – 7 = 9

9 – 9 = 0

Since the process is repeated 5 times, therefore, we have,√25 = 5.

Question 6. Compute the square root of 484 by the long division method?

By the long division method, we have,

The remainder is 0, therefore, 484 is a perfect square number, such that,

√484 = 22

Define a 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.

Types of Square Root Methods

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.

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.

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.

5. The value of the quotient is doubled and entered in the blank space on the right side.

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.

Square Root Tricks

The square root is a complex operation. Hence, it can be a time-consuming operation to obtain the square roots of a number. There are small tricks that can be used to determine the square root of a number faster. Stay tuned to learn such tricks and see the speed of your calculation increase!!!

The square root of a number is the number that on squaring results the given number. It is represented using this symbol «√». The square root of a number can be a rational number or an irrational number. If the square root of a number is a whole number, then it is a perfect square. In order to calculate the square root of a number easily, square root tricks are used. Let’s find out more about it.

Tricks to Calculate Square Root

We can estimate the square root of a perfect square number using a trick. In order to determine the square root without using a long division, we must know the unit digits of squares of the first ten numbers.

It is possible to calculate the square root of a number having 4 digits as well as 5 digits. Let us learn to calculate the square root of numbers using the trick.

Square Root Trick for 4 Digit Numbers

The steps to estimate the square root of a 4 digit number are:

Steps to find units digit if the unit digit obtained in Step 2 is apart from 5 or 10 are:

Example: Find the square root of 1521.
Solution: The steps to determine the square root of 1521 are:

Square Root Trick for 5 Digit Numbers

The steps to estimate the square root of a 5 digit number are:

Example: Find the square root of 10816.
Solution: The steps to determine the square root of 10816 are:

Topics Related to Square Root Tricks

The square root is an arithmetic operation represented by a symbol √ and its value is the number that on squaring gets results from the given number. For example, square root of 4 is 2 as √4 = √(22) = 2.

What is the Trick to Find Square Root?

The square root of a number can be found by using the below steps:

How Do You Find the Square Root of a 3 Digit Number Using Square Root Tricks?

The square root of a 3 digit number is always a 2 digit number. Hence in order to understand using square root tricks for a 3 digit number, let us consider taking the square root of 324.

Thus, the square root of 324 is 18.

What is the Easiest Way to Find the Square Root of a 4 Digit Number?

The Easiest Way to Find the Square Root of a 4 Digit Number is by using square root tricks.

Is the Square root of 169 a Whole Number?

Yes, the square root of 169 is a whole number as √169 = 13.

First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number: multiply it by itself.

What is 3 squared?

«Squared» is often written as a little 2 like this:

This says «4 Squared equals 16″(the little 2 says
the number appears twice in multiplying)

Squares From to

We can also square negative numbers.

That was interesting!

When we square a negative number we get a positive result.

Just the same as squaring a positive number:

A square root goes the other way:

3 squared is 9, so a square root
of 9 is 3

It is like asking:

What can we multiply by itself to get this?

Here are some more squares and square roots:

Decimal Numbers

It also works for decimal numbers.

Using the sliders:

• What is the square root of 8?
• What is the square root of 9?
• What is the square root of 10?

Negatives

We discovered earlier that we can square negative numbers:

(−3) squared

(−3) × (−3) = 9

And of course 3 × 3 = 9 also.

So the square root of 9 could be −3 or +3

What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

So the square roots of 25 are −5 and +5

The Square Root Symbol

We use it like this:

What is 25?

25 = 5 × 5, in other words when we multiply
5 by itself (5 × 5) we get 25

So the answer is:

But wait a minute! Can’t the square root also be −5? Because (−5) × (−5) = 25 too.

What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The Perfect Squares (also called «Square Numbers») are the squares of the integers:

Try to remember them up to 12.

Calculating Square Roots

It is easy to work out the square root of a perfect square, but it
is really hard to work out other square roots.

What is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:

But the digits just go on and on, without any pattern.

So even
the calculator’s answer is only an approximation !

Note: numbers like that are called Irrational Numbers, if you want to know more.

The Easiest Way to Calculate a Square Root

And also use your common sense to make sure you have the right answer.

A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

3.1623 x 3.1623 = 10.00014

How to Guess

In that case we could think «82,163» has 5 digits, so the square root might have 3 digits (100×100=10,000), and the square root of 8 (the first digit) is about 3 (3×3=9), so 300 is a good start.

Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like 4/4/16

The next after that is the 5th of May 2025 (5/5/25)

309,310,315, 1082, 1083, 2040, 3156, 2041, 2042, 3154

You might like to read our Introduction to Squares and Square Roots first.

This says «4 Squared equals 16″(the little 2 means
the number appears twice in multiplying, so 4×4=16)

A square root goes the other direction:

What can I multiply by itself to get this?

Here is the definition:

A square root of x is a number r whose square is x:

r2 = x
r is a square root of x

We can use it like this:

we say «square root of 9 equals 3»

It isn’t clear! And we get different answers:

So let’s make it clear by using «( )».

Just the same as when we square a positive number:

Now remember our definition of a square root?

And we just found that:

(+5)2 = 25
(−5)2 = 25

So both +5 and −5 are square roots of 25

Two Square Roots

There can be a positive and negative square root!

This is important to remember.

Solve w2 = a

w = √a   and   w = −√a

Principal Square Root

So if there are really two square roots, why do people say √ =

There are two square roots, but the symbol means just the principal square root.

The square roots of 36 are 6 and −6

But √ = (not −6)

The Principal Square Root is sometimes called the Positive Square Root (but it can be zero).

In a Nutshell

When we have:r2 = x

r = ±√x

Why Is This Important?

Why is this «plus or minus» important? Because we don’t want to miss a solution!

Solve x2 − 9 = 0

x2 − 9 = 0

Move 9 to right:x2 = 9

x = ±√9

x = ±3

The «» tells us to include the «−3» answer also.

Solve for x in (x − 3)2 = 16

(x − 3)2 = 16

x − 3 = ±√16

Add 3 to both sides:x = 3 ± 4

x = 7 or −1

Check: (7−3)2 = 42 = 16
Check: (−1−3)2 = (−4)2 = 16

Square Root of xy

When two numbers are multiplied within a square root, we can split it into a multiplication of two square roots like this:

√ = √√

but only when x and y are both greater than or equal to 0

What is √(100×4) ?

= √(100) × √(4)

= 10 × 2

And √√ = √ :

What is √8√2 ?

√(−8 × −2) = √(−8) × √(−2)

We seem to have fallen into some trap here!

We can use Imaginary Numbers,

but that leads to a wrong answer of −4

The rule only works when x and y are both greater than or equal to 0

So we can’t use that rule here.

Instead just do it this way:

√(−8 × −2) = √16 = +4

Why does √ = √√ ?

We can use the fact that squaring a square root gives us the original value back again:

(√)2 = a

Assuming a is not negative!

We can do that for xy:(√)2 = xy

And also to x, and y, separately:(√)2 = (√)2(√)2

Use a2b2 = (ab)2:(√)2 = (√√)2

Remove square from both sides:√ = √√

An Exponent of a Half

A square root can also be written as a fractional exponent of one-half:

but only for x greater than or equal to 0