Formula and Execution

Number

In this section, you will type the number whose square root is required.

Square Root

This section calculates the square root of the numbers you have typed in the above section and gives you the required results.

Square Root Calculator

Almost most of your math homework problems require you to calculate the square root of a certain number. Most of us know the basic squares like √4=2, √9=3, √25=5, but when it comes to the higher numbers or the numbers that are not the perfect squares, then it gets difficult and time-consuming to carry out the entire square root calculating process.

So, you do not have to worry about it anymore because we have taken care of that for you. Just type in your number in our online square root calculator’s Number section and get results in a heartbeat!

How to Calculate Square Root Using Calculator Beast Square Root Calculator

It is as simple as you could possibly think. A screenshot of the calculator shows an example of the square root of 4578.

Square Root Calculator

Our square root calculator is smart enough to work both ways. It not only calculates the square root of any number, but it also generates squares in the number section if you substitute the value in the square root section. A screenshot of the example shows the square of the square root of 27.56.

Square Root Calculator

Concept of Square Root

Ahan! We decided to bring for you, not just the finest online square root calculator, but also to make you learn every possible thing about the square roots: their history, their concept, their properties, and much more about them. So read further to get some useful insights about square roots.

Squares

Before we move on to square roots, let us learn what squares are and why they are called squares.

We have studied exponents and powers, and we know that any number or variable which is raised to the power of 2, is known as a square. Like 22 is read as 2 squared and x2 is read as x squared. 

Have you ever thought why exactly we have names for just squares and cubes and not for the powers above 3? Like x3 is read as x cubed while x4 is read as x raised to 4.

To know about why we have the term square for exponent 2, read further. And to know about the term cube, check our online cube root calculator.

Squaring a number means multiplying the number by itself. Like if we take 32, we are multiplying 3 by itself and the result is 9. Mathematically, we write it as,

32 = 3*3 = 9

Now, if we plot this on a graph, taking one 3 in the horizontal axis, and the other in the vertical axis as shown, we get a square which has an area of 9 square units. This is where the word square comes from when the number is multiplied to itself.

Squares

Square Roots

Now just as we have the subtraction operation as the inverse of addition, the division operation as the inverse of multiplication, we have square roots as the inverse of squares.

Like in the example above, we saw that the square of 3 is 9. What if we have to calculate the square root of 9? Well, the answer would be 3, because the square root is the inverse of the square, which means it will lead you to the original number that generated 9 when squared.

The symbol we use for square roots is a radical sign, ‘√’ which is sometimes written with a bar ‘√‾’ over the numbers. 

The origin of this radical sign is not exactly known, but it is said that it comes from the letter ‘r’, which in Greek or Latin is used for Radixwhich is their first letter meaning root or base. So the word radix might be interpreted for the base of the square. And maybe the word radical for the symbol of square root comes from radix too, who knows?

If we define square roots mathematically, we say.

“If a number y can be expressed as y = x2, we say that y is the square of and x and x is the square root of y.”

y = x ó y = x2

Examples:

Some examples are given to develop a deep understanding of squares and square roots.

  1. Find the square root of (a) 529, (b) 2025

Solution

To find the value of the square root of 529, substitute 529 in the number section of our square root calculator to get 23 as a result.

Similarly, substitute 2025, and the result is 45.

  1. Find the square of (a) 35, (b) 67

Solution

Our calculator works both ways. It not only gives you the value of the square root of the numbers but even generates squares from the square roots. 

Substitute 35 in the square root section of the calculator, and you will get 1225 as a result in the number section of the calculator.

Similarly, when you substitute 67 in the square root section and the answer will be 4489.

Perfect Squares

To develop the understanding of a perfect square, we follow examples.

The square root of 16 is 4, while the square root of 20 is 4.472. 

So when the square root is a natural number, it is known as the ‘perfect square’. In the example above, 16 is the perfect square of 4. While if the square root is in decimal, it is not a ‘perfect square’. 20 is not the perfect square of any number because its square root is irrational.

Note: We do not use the term ‘imperfect squares’ for decimal square roots.

Some Commonly Used Values of Square Root

Below are the most commonly used squares and square roots that, despite using the calculator, one must learn.

xx2x√x
1111
2442
3993
416164
525255
636366
749497
864648
981819
1010010010
1112112111
1214414412
1316916913
1419619614
1522522515
1625625616
1728928917
1832432418
1936136119
2040040020

Facts About Squares and Square Roots

Below are some facts about the square root that you may find interesting.

  1. Even numbers have even squares, while odd numbers have odd squares.
    E.g.: 22 = 4, and 32 = 9
  2. Numbers having digits 2, 3, 7, and 8 in the units place will never be a perfect square.
    E.g.: Just by looking at these numbers 5672, 4563, 987, 8088, we can conclude that they are not the perfect squares.
  3. From statement 2, it would be ridiculous to conclude that all numbers having 0, 1, 4, 5, 6, and 9 in the units place would be perfect squares. They may and may not be a perfect square.
    E.g.: 196 has 6 in the units place, and it is a perfect square of 14. But 286 also has 6 in the units place, but it is not a perfect square of any number. 
  4. If a number contains an odd number of zeroes, in the end, it will never be a perfect square. But if it ends in an even number of zeroes, it may or may not be a perfect square.
    E.g.: 1000 has 3 zeroes in the end, it will never be a perfect square. While 10000 has 4 zeroes in the end and it is a perfect square of 100, while 20000 also ends with an even number of zeroes, but it is not a perfect square of any number.
  5. To find the number of digits in the square roots, we have a very basic formula. If the number of digits of the number is even, we calculate the number of digits of the square root by the formula n/2. And if the number of digits of the number is odd, then the formula is (n+2)/2
    E.g.: If we want to know the number of digits of the square root of 9025, we simply divide 4 by 2. So our answer would be 2 digits. And when we calculate the square root of 9025, the result is 95 and we can see that these are two digits.
    Now, I leave for you to calculate the number of digits of square root for the following number: 11236

How to calculate Square Root by Division Method?

To learn how to calculate the square root by division method, I assume you know prime factorization. If you are unaware of the concept or you think you need a quick revision, check out our GCF calculator.

We consider 2916 to calculate the square root by division method. We simply use the prime factorization to calculate the factors of 2916.

The prime factors of 2916 are: 2, 2, 3, 3, 3, 3, 3, and 3

We write the above thing mathematically as,

√2916 = √ (2 x 2 x 3 x 3 x 3 x 3 x 3 x 3)

            = √{(2x2)(3 x3)(3 x 3)}                            (Pairing the above quantities)

We can write the above equation as 

√2916 = √(2 x 3 x 3 x 3)2

Since we already discussed that the square and square root are inverses of each other, so they both cancel out, and we are left with,

√2916 = 2 x 3 x 3 x 3

            = 54

So the square root of 2916 is 54. Now verify this by substituting 2916 in the number section of the calculator at the left-hand side, and you will see 54 as a result!

How to estimate the value of Squares and Square Roots?

Estimation of squares and square roots is usually required in exams when the calculator is not allowed, and besides, it is also a fun thing to do if you are a math enthusiast. So the values we estimate are usually of numbers not bigger than 2 to 3 digits.

To estimate the values mentally, it is important to learn the table given above and also learn the short trick of finding the squares of the numbers that end in 0 and 5.

If a number ends in 0 like 30, simply multiply 3 by 3 and double the number of zeroes. The result is 900. I hope you can easily tell the square root of 140? Well, 14 into 13 gives 196, and when we double the zeroes, we get 19600 as the square of 14.

Now we learn to calculate the square of the numbers ending in 5. Remember one thing, the square of the number that has 5 in the end, will always have 25 in the end. What you have to do is simply multiply the first digit of the number by the immediate next number.

Like, consider the number 35. We take first digit 3 and multiply it by the immediate next number 4. The answer is 12. Now simply write 25 after 12. So the square of 35 is 1225.

Now calculate for 45. What did you get? 2025? Perfect! Easy-peasy, right?

So well, now I assume that you have memorized the squares and square roots till 20 (the table above), and also, the above short tricks are clear to you. We move on to learning the estimation of squares and square roots.

To estimate the value of 372, we consider 402 and 352 as the reference point. Since 352 = 1225 and 402 = 1600, so the value of 372 would be somewhere between 1225 and 1600.

To estimate the value of √63, we recall the square roots we learned. We know √64 = 8, the √63 would be somewhat less than 8. 

These mental estimations are easy when you know the table and the short tricks.

How to Calculate Square Root Geometrically?

Yes, you read it right! Geometrically! Interesting, isn’t it? Well, as interesting it is, as easy it is.

Suppose you want to calculate the square root of x. Draw a number line AC and mark x+1 units on that line. Why x+1? Because it works that way. No matter whatever be the number, it always works this way. Now divide the number line by 2 to get the center O. With center as radius, draw an arc from A to C. A semi-circle is obtained. Now point B is the length of x. from this point, draw a perpendicular line that touches the semi-circle at point D as shown in the figure.

How to Calculate Square Root Geometrically

Now measure the length of the line BD. The answer will be the square root of x. Impressive, right? Well, now go on and play with whatever values you wish to. And guess what? It works even for numbers that aren’t perfect squares. A slight error would exist, but you can great results up to one decimal place if the figure is flawlessly drawn.

Simplifying Equations with Square Roots

Simplifying equations involving square roots is not that big a deal if you know how to simplify algebraic equations.

For example, if you have the equation 2x + 3y + 4z - x + 5y - 7z = 0, you would bring together all the similar variables in one place and simplify them up. Your final result will be x + 8y – 3z = 0.

The case with square roots is exactly the same. √a will never be added to or subtracted from √b. They are treated exactly the way we treated x, y, and z in the above equation. 

Consider the following example,

3√2 + 7√5 -4√3 = 4√2 - 8√3 + 3√5

Now consider √2, √3, and √5 as separate variables and rearrange the equation accordingly

  • (3√2 -4√2) + (7√5 - 3√5) + (-4√3 + 8√3) = 0
  • -√2 + 4√5 + 4√3 = 0

That wasn’t difficult, was it? Just remember never to make a mistake like this one: √2 + √3 = √5. This would be so unfair to mathematics!

We just discussed the addition and subtraction operations with the equations involving square roots. What if we have to perform multiplication and division? Well, that again is a no big deal, and we deal with it exactly the way we deal with algebraic equations. Like 

x*y = xy, 

So √2 * √3 = √(2*3) = √6

Similarly, √4 / √8 = √(4/8) = √(1/2)

Eliminating Square Roots

Eliminating square roots is, again, no rocket science. We know that squares are the inverse of square roots. So to eliminate square root from any equation, we simply square both sides of the equation.

Example

√x = 3 + x

Squaring both sides of the above equation will yield

(x) = (3 + x)2

Since squares and square roots are inverses, they cancel out each other, and we are left with,

x = (3 + x)2

Solving the above equation will give us a quadratic equation. That can be further solved by factorization or quadratic formula without any hassles to deal with the square root.

Graph of a Square Root Function

A function is nothing but a machine that generates an output when you put input into it. A square root function is a function that contains a variable with a radical sign. The simplest example of a square root function is,

F(x) = 

When we substitute values for x, we get values for f(x). Here f(x) is read as ‘f of x’.

For 1, f(1) = √1 = 1

For 2, f(2) = √2 = 1.414

For 3, f(3) = √3 = 1.73

For 4, f(4) = √4 = 2

And this can go on forever.

When we plot these values, we get a graph that is half parabola, as shown below.

Graph of a Square Root Function

Now when we notice the graph, we see that it is continuous, growing, and its limit tends to infinity. The graph also changes at every point, so its rate of change, i.e., derivative, can be calculated. So we say that the square root is a differentiable function

The Derivative of a Square Root

A derivative is nothing but the rate at which the function changes with respect to time. As we saw above that the derivative of the square root is possible as it is a continuously changing function.

The simplest example is the distance function. We know that the rate of change of distance is velocity. We write distance function as x(t) and velocity function as v(t) since they both change with respect to time. The derivative of x(t) is written by placing a “” over x like written below.

v(t) = x’(t)

The basic formula to calculate the derivative of a function is 

f(x) = xn ó f’(x) = nx(n-1)

Example: The derivative of 2x3 is 6x2

When we calculate the derivative of square root function, we consider the fact that we replace the √ sign by ½ in power, because the square root is the inverse of the square and so is ½ of 2.

f(x) = √x

f(x) = (x)1/2 

The derivative would be,

f’(x) = (1/2) x1/2 – 1 

f’(x) = (1/2) x-1/2 

f’(x) = 1/(2x1/2) 

f’(x) = 1/(2√x)

FAQs

How do I calculate the square root by calculator?

Just find the button that has a ‘√’ sign, press it, and enter the value whose square root is required, press = sign, and you are all good to go!

How many possible values do we get when we calculate the square root?

The answer is not 1. Amazed? Well, let me explain. When we multiply a positive number by a positive number, we get a positive number, and when we multiply a negative number by a negative number, we again get a positive number. Like,

5*5 = 25

And also, -5*-5 = 25

So when we calculate √25, we do not know whether this 25 was the result of two positive 5s or two negative 5s. This can be written in both ways as,

√25 = √(5*5)

And, √25 = √(-5*-5)

So, our answer can both be 5 and -5. To avoid this confusion, we always consider two possible values of square roots, and we write it as,

√25 = ± 5

So, the answer to the question is, we get two possible values when we calculate the square root of any number.

What is the square root of a negative number?

Well, if you are in school, then the square root of negative numbers can probably take you to the world of numbers you would’ve never heard before. The world of ‘complex numbers’. So what it is, let us dive a bit deep into it.

What do you think is the √(-25)? -25 is the product of one positive five and one negative 5, i.e., -25 = 5*-5

Remember that 5 and -5 are two separate numbers, and they cannot be considered the same. They exist at different places on a number line or a Cartesian plane. So, this leads us to the problem of calculating the square root of negative numbers.

Mathematicians devoted their lives to this cause until they found a solution to it. And the solution was to write the negative number as the multiple as -1, i.e.,

√(-25) = √{(-1)(5*5)}

This gave 

√(-25) = √(-1) √(5*5)

            = ± 5√-1

They called this √-1 on a Greek symbol iota and which is written as “i”.

So,

 √(-25) = ± 5i

Now, calculate the square root of the following to test how much clear are you.

  • √(-64)
  • √(-196)
  • √(-6561)

What are rational and irrational square roots?

They are exactly what the rational and irrational numbers are. Some square roots can be written in fraction form, like √64 = 8/1, these are rational square roots. While some square roots cannot be written in the form of a fraction, like, √2 = 1.14142113…, these are known as irrational square roots.

Do decimal numbers have square roots? 

The simple answer is, why not? They do have square roots. The first step is to convert the decimal into a fraction. Now calculate separate square roots of numerator and denominator and then divide them both to get the final answer.

Example:

√1.25 = √(125/100) = √125/√100 = 5√5/10 = √5/2 = 1.118

Fun Fact!

The old Greeks believed that the numbers could only be natural or rational. They did not have the concept of irrational numbers. They believed that every number could be written in the form of a fraction; thus, every number is rational. Hippasus, the student of Pythagoras, was the first man who calculated the square root of 2 and found out that it cannot be expressed in the form of fractions. This hurt the highfalutin of the Greeks so much that Hippasus was never seen after that. He is believed to be drowned as punishment. 

Ah! Poor guy! If he was alive today, he would be so proud of himself.