Square Root Tricks

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

In mathematics, a square root of a number x is such that, a number y is the square of x, simplify written as y2 = x.

, 5 and – 5 are both square roots of 25 because:

5 x 5 = 25 and -5 x -5 =25.

The square root of a number x is denoted with a radical sign √x or x 1/2. For instance, the square root of 16 is presented as: √16 = 4. A number whose square root is calculated is referred to as radicand. In this expression, √16 = 4 number 16 is the radicand.

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.

The principal square root is the one with a positive real part. The only case when the formula fails is when there is no principal square root, that is, when is a negative real number.

Here’s the picture for \$z = 9 + 4i\$:

Remark: The construction of the square roots is geometrically exact. That is to say, they were constructed using straightedge and compass only. I decided to hide the construction, as it seems rather obfuscating the intended illustration than adding to it. Nevertheless, I suggest taking a moment and thinking about how you would achieve the geometric construction.

The pictures were created using GeoGebra.

We say \$x\$ is a «square root» of \$y\$ if \$x^2=y\$. Thus, both \$+7\$ and \$-7\$ are square roots of \$49\$.

The same kind of «having to make a choice» situation arises if one wants to define a square root function for complex numbers. We can no long impose the same kind of continuity conditions and get a straight answer — instead we have to form a sort of «barricade» in which the value of the square root jumps dramatically when we cross over this barricade. This is known as a branch cut.

The idea of branch cuts leads into more advanced complex analysis topics of monodromy (which pertains to «running around» a singularity, like crossing over the branch mentioned earlier) and also Riemann surfaces, which can be thought of as what we get when we refuse to cut the plane into branches and instead consider a function multi-valued and look at its graph (I am probably butchering that description though).

Need to find the square of a number? Finding a number’s square is as simple as multiplying it by itself. To square fractions, find the squares of both the numerator and denominator. Then reduce or simplify the result. Keep reading for helpful tips and examples!

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Asking what a square root «is», to a mathematician’s ear, means asking how it is defined: if I may read into what you’re saying, that’s not what you’re trying to figure out here. But knowing how some other things are defined will help.

So, let’s start with the real numbers: what you know as streams of decimal digits are generally defined in some way or another as all of the spaces between rational numbers. Rational numbers are the whole-number-fractions that you learned about in school. If you want to read up more on how to formally do this, those spaces are called Dedekind cuts.

There are two other key observations about squaring numbers which makes it easier to produce a square root. The first one is: if \$0 < a < b\$, then \$a^2 < b^2\$, this has one of those scary math-names (monotonic). The second one is that if two rationals are near each other, then their squares are near each other (continuous). (This latter property is actually even better because squaring is not just continuous but differentiable, which is a complicated word that means that, when you zoom in on the graph, it becomes a straight line everywhere — it’s like a circle, not like a fractal.)

Since it’s continuous, the square root of any positive real number is always a well-defined positive real number: Given the positive gap-between-rationals which you want to take the square root of, the square root is the positive gap-between-rationals such that any rational greater than the square-root-gap squares to a rational greater than the square-gap, similarly for numbers below the gaps. So every real number has a real square root.

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:

Square Roots

A square root goes the other way:

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

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

√25 = 5

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.

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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

Square root is one of the most important functions in Mathematics which has a wide range of applications in day to day life and also scientific calculations. Square root of any number in Mathematics is that number which when multiplied by itself gives the product equal to the number whose square root is to be determined. Square root of a number is represented as the number written within the symbol ‘√’. Square root of a number ‘x’ is written as √x. Square root of a number can be represented in exponential form as the number to the power ½. Square root of a number ‘x’ can be written in exponential form as (x)

What is a Perfect Square Number and Square Root Formula?

It is very important to understand what is a square root of a perfect square number before making yourself clear about what is a root in math. What is a perfect square number in Mathematics can be that number which is obtained as a product by multiplying any integer by itself. The square root formula when used for perfect square numbers will yield a number which is an integer as the answer. i.e. Square root of a perfect square number is always an integer.

What is a Root in Math?

There are several methods to find the square root of a number among which a few familiar ones are:

Finding Square Root Formula by Prime Factorization Method

Prime factorization method is a method in which the numbers are expressed as a product of their prime factors. The identical prime factors are paired and the product of one element from each pair gives the square root of the number. This method can also be used to find whether a number is a perfect square or not. However, this method cannot be used to find the square root of decimal numbers which are not perfect squares.

Evaluate the root of 576.

So, 576 can be written as a product of prime numbers as:

576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3

Square root of 576 = 2 x 2 x 2 x 3 = 24

Square Root Formula Using Repeated Subtraction Method

This is a method in which the number whose square root is to be determined is repeatedly subtracted by the consecutive odd number till the difference becomes zero. The number of subtractions gives the root of the number.This method can only be used to find the square root of perfect square numbers.

Estimate the Square root of 16

The number is subtracted from odd numbers starting from 1.

16 — 1 = 15

15 — 3 = 12

12 — 5 = 7

7 — 7 = 0

Number of subtractions here is 4. So, the square root of 16 is 4.

Average Method of Square Root Formula

In this method, the concept of average is used to find the square root of a given decimal number. This method can be conveniently used to find the square root of whole numbers upto a few decimal places.

Evaluate the square root of 3 using the average method.

The two square numbers in between which ;3’ lies are 1 and 4. So, the square root of 3 lies between 1 and 2. Find the average of these two numbers to get the square root of 3.

Square root of 3 = (1 + 2)/ 2 = 3/ 2 = 1.5 which is not accurate. So, finding the average is further continued as

Square root of 3 = (1.5 + 2)/2 = 1.75 which is approximately equal to square root of 3.

Fun Facts About Square Root Formula

The article has presented complete insight about the Square Root formula that will help students to practice and learn.

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 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

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.

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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.

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.

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

Given below is the step by step method:

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

In the days before calculators, students and professors alike had to calculate square roots by hand. Several different methods have evolved for tackling this daunting process, some giving a rough approximation, others giving an exact value. To learn how to find a number’s square root using only simple operations, please see Step 1 below to get started.

Using a Long Division Algorithm

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.

Example

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).

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

= √(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,

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:√ = √√

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?

Negative Numbers

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?

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

r2 = x
r is a square root of x

And we just found that:

(+5)2 = 25
(−5)2 = 25

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

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

To find the square of a number, multiply the number by itself. For example, if you’re trying to find the square of 5, you would multiply 5 by 5 and get 25, which is the square. To learn how to square fractions, scroll down!

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In a Nutshell

When we have:r2 = x

r = ±√x

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

What is Square of a Number?

In mathematics, square of a number is the result of multiplying the number by itself. The word square is usually equivalent to raising a number to the power of 2 and denoted by the superscript 2.

, the square of 4 is written as 42 which gives 16 as the answer. In this case, 16 is the square of number 4.

Below is a list of squares of the first twelve numbers:

1 x 1 = 1                           7 x 7 = 492 x 2 = 4                           8 x 8 = 643 x 3 = 9                           9 x 9 = 814 x 4 = 16                        10 x 10 = 1005 x 5 = 25                        11 x 11 = 1216 x 6 = 36                        12 x 12 = 144

Squaring Negative Numbers

The square of a negative number is a positive number. For instance, -3 x -3 would become 9, however – 3 x 3 = -9, this is because -3 is a different number to 3.

We can use it like this:

we say «square root of 9 equals 3»

There are multiple ways to find the square of the numbers. We will see a few of the here.

This method involves, successful and repeated subtraction of odd numbers such as 1, 3, 5 and 7 from the number until zero is reached. The square of the number is equal to the number or frequency of subtraction performed on the number

Suppose, we need to calculate the square of a perfect number like 25, the operation is done as:

You can notice that, the frequency of subtraction is 5, therefore the square root of 25 is 5.

In this method, a perfect square number is factorized by successive division. The prime factors are grouped into pairs, and the product of each number calculated. The product is therefore, the square root of the number. To find the square of a perfect number such as: 144 is performed as:

The square root of 225 is calculated as

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

To find a square root of a number without a calculator, see if you can get to that whole number by squaring smaller numbers, or multiplying a smaller number by itself. If the number is a perfect square, you will get a whole number as the square root. Otherwise, try squaring numbers with a decimal until you get as close as possible to your original number. If you want to learn how to estimate the square root of imperfect squares, keep reading the article!

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What is Square Root of a Number?

The square root is an inverse operation of squaring a number. In other words, the square root is an operation that undoes an exponent of 2. A square root of a number x is such that a number y is the square of x, simplify written as y2 = x.

The square root of a number x is denoted with a radical sign √x or x 1/2. For instance, the square root of 16 is represented as √16 = 4. A number whose square root is calculated is referred to as radicand. In this expression, √16 = 4, number 16 is the radicand.

Finding the Square Root of Numbers

I am thinking of two numbers. Both numbers are square numbers greater than \$1\$. If the sum of these numbers is \$100\$, what are the values of the two numbers?

\$6\$ and \$8\$

\$8\$ and \$12\$

\$16\$ and \$84\$

\$36\$ and \$64\$

True or False: There are nine perfect squares between \$0\$ and \$100\$.

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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.

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.

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.

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

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

Square root is an inverse operation of the squaring a number. In other words, square root is an operation that undoes an exponent of 2.

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