Our square root calculator estimates the square root of any positive number you want. Just enter the chosen number and read the results. Everything is calculated quickly and automatically! With this tool, you can also estimate the square of the desired number (just enter the value into the second field) which may be a great help in finding perfect squares from the square root formula. Are you struggling with the basic arithmetic operations: adding square roots, subtracting square roots, multiplying square roots or dividing square roots? No more! In the following text, you will find a detailed explanation about different square roots properties, e.g., how to simplify square roots, with many various examples given. With this article, you will learn once for all how to find square root!

Have you ever wondered what is the origin of the square root symbol √? We can ensure you that this history is not as simple as you could think at first. The origin of the root symbol goes back to ancient times, as the origin of the percent sign.

If you're looking for the square root graph or square root function properties, head directly to the appropriate sections. There, we explain what is the derivative of square root using fundamental square root definition; we also elaborate on how to calculate square roots of exponents or square roots of fractions. Finally, if you are persistent enough, you will find out that square root of a negative number is, in fact, possible. In that way, we introduce complex numbers which find broad applications in physics and mathematics.

Square root symbol √

The operation of the square root of a number was already known in antiquity. The earliest clay tablet with the correct value of up to 5 decimal places of √2 = 1.41421 comes from Babylonia (1800 BC - 1600 BC). Many other documents show that square roots were also extracted by ancient Egyptians, Indians, Greeks, and Chinese. However, the origin of the root symbol √ is still largely speculative.

  • many scholars believe that square roots originate from the letter "r" - the first letter of the Latin word radix meaning root,
  • another theory states that square root symbol was taken from the Arabic letter ج that was placed in its original form of ﺟ in the word جذر - root (the Arabic language is written from right to left).

The first use of the square root symbol √ didn't include the horizontal "bar" over the numbers inside the radical symbol √‾, known as a Latin vinculum which means bond. Although radical symbol with vinculum is in the everyday use, we usually omit this overline in the many texts, like in the articles on the internet sites. The notation of the higher degrees of a root has been suggested by Albert Girard who placed the degree index within the opening of the radical signs, e.g., ∛ or ∜.

The last question is why the square root operation is called root regardless of its true origin? The explanation should become more evident if we write the equation x = ⁿ√a in a different form: xⁿ = a. Quantity x is called root or radical because it is the hidden base of a. Thus, the word radical doesn't mean far-reaching or extreme, but foundational, reaching to the root causes.

Square root definition

In mathematics, the traditional operations on numbers are addition, subtraction, multiplication, and division. Nonetheless, we sometimes add to this list some more advanced operations such as manipulations square roots, exponentiation, logarithmic functions and even trigonometric functions (e.g., sine and cosine). In this article, we will focus on the square root definition only.

The square root of a given number x is every number y whose square y² = y*y yields the original number x. Therefore, the square root formula can be expressed as:

√x = y ⟺ x = y²,

where is a mathematical symbol that means if and only if. Each positive real number always has two square roots - the first is positive and second is negative. However, for many practical purposes, we usually use the positive one. The only number that has one square root is zero. It is because √0 = 0 and zero is neither positive nor negative.

There is also another common notation of square roots that could be more convenient in many complicated calculations. This alternative square root formula states that square root of a number is a number raised to the exponent of the fraction one half:

√x = x^(1/2) = x^(0.5)

In geometric interpretation, the square root of a given area of a square gives the length of its side. That's why has word square in its name. A similar situation is with the cube root . If you take the cube root of a volume of the cube, you get the length of its edge. While square roots are used when considering surface areas, cube roots are useful to determine quantities that relate to the volume, e.g., density.

How to find the square root?

Maybe we aren't very modest, but we think that the best answer to the question how to find the square root is straightforward: use the square root calculator! You can use it both on your computer and your smartphone to quickly estimate the square root of a given number. Unfortunately, there are sometimes situations when you can rely only on yourself, what then? To prepare for this, you should remember several basic perfect square roots:

  • square root of 1: √1 = 1, since 1 * 1 = 1;
  • square root of 4: √4 = 2, since 2 * 2 = 4;
  • square root of 9: √9 = 3, since 3 * 3 = 9;
  • square root of 16: √16 = 4, since 4 * 4 = 16;
  • square root of 25: √25 = 5, since 5 * 5 = 25;
  • square root of 36: √36 = 6, since 6 * 6 = 36;
  • square root of 49: √49 = 7, since 7 * 7 = 49;
  • square root of 64: √64 = 8, since 8 * 8 = 64;
  • square root of 81: √81 = 9, since 9 * 9 = 81;
  • square root of 100: √100 = 10, since 10 * 10 = 100;
  • square root of 121: √121 = 11, since 11 * 11 = 121;
  • square root of 144: √144 = 12, since 12 * 12 = 144;

Above numbers are the simplest square roots because every time you obtain an integer. Try to remember them! But what can you do when there is a number that doesn't have such a nice square root? There are multiple solutions. First of all, you can try to predict the result by trial and error. Let's say that you want to estimate what is the square root of 52:

  1. You know that √49 = 7 and √64 = 8 so √52 should be between 7 and 8.
  2. Number 52 is closer to the 49 (effectively closer to the 7) so you can try guessing that √52 is 7.3.
  3. Then, you square 7.3 obtaining 7.3² = 53.29 (as the square root formula says) which is higher than 52. You have to try with a smaller number, let's say 7.2.
  4. The square of 7.2 is 51.84. Now you have a smaller number, but much closer to the 52. If that accuracy satisfies you, you can end estimations here. Otherwise, you can repeat the procedure with a number chosen between 7.2 and 7.3,e.g., 7.22 and so on.

Another approach is to simplify square root first and then use the approximations of the prime numbers square roots (typically rounded to the two decimal places):

  • square root of 2: √2 ≈ 1.41,
  • square root of 3: √3 ≈ 1.73,
  • square root of 5: √5 ≈ 2.24,
  • square root of 7: √7 ≈ 2.65,
  • square root of 11: √11 ≈ 3.32,
  • square root of 13: √13 ≈ 3.61,
  • square root of 17: √17 ≈ 4.12,
  • square root of 19: √19 ≈ 4.34, etc.

For example, what is the square root of 52? You can simplify it to the √52 = 2√13 (you will learn how to simplify square root in the next section) and then substitute √13 ≈ 3.61. Finally, make a multiplication √52 ≈ 2 * 3.61 = 7.22. The result is the same as before!

You can check whether a number is prime or not with our prime number calculator. A prime number is a natural number (greater than one) that can't be obtained as a product of two smaller natural numbers. For example, 7 is a prime number because you can get it only by multiplying 1 * 7 or 7 * 1. On the other hand, number 8 is not prime, because you can form it by multiplying 2 * 4 or 4 * 2 (besides product of 1 and 8 itself).

Square root calculator

In some situations, you don't need to know the exact result of the square root. If this is your case, our square root calculator is the best option to estimate the value of every square root you desired. For example, let's say you want to know whether 4√5 is greater than 9. From the calculator, you know that √5 ≈ 2.23607, so 4√5 ≈ 4 * 2.23607 = 8.94428. It is very close to the 9, but it isn't greater than this! The square root calculator gives the final value with relatively high accuracy (to five digits in above example). With the sig fig calculator, you can calculate this result to as many significant figures as you want.

Remember that our calculator automatically recalculates numbers entered in any of the two fields. You can find what is a square root of a specific number by filling the first window or get the square of a number that you entered in the second window. The second option is handy in finding perfect squares that are essential in many aspects of math and science. For example, if you enter 17 in the second field, you will find out that 289 is a perfect square.

In some applications of the square root, particularly those pertaining to sciences such as chemistry and physics, the results are preferred in scientific notation. In brief, an answer in scientific notation must have a point between the first two non-zero numbers and will be represented as the decimal multiplied by 10 raised to an exponent. For example, the number 0.00345 is written as 3.45 * 10⁻³ in scientific notation, whereas 145.67 is written as 1.4567 * 10² in scientific notation. The results obtained using the square root calculator can be converted to scientific notation with the scientific notation calculator.

How to simplify square roots?

At first, let's ask a question which square roots can be simplified. To answer it, you need to take a number which is underneath the square root symbol and find its factors. If any of its factors is 4, 9, 16, 25, 36, 49, 64 and so on, then you can simplify the square root. What is special in those numbers? They can be respectively expressed as 2², 3², 4², 5², 6², 7² and so on. According to the square root definition, you can call them perfect squares. We've got a special tool called factor calculator which might be very handy there. Let's take a look at some examples:

  • can you simplify √27? With the calculator mentioned above, you obtain factors of 27: 1, 3, 9, 27. There is 9 here! This means you can simplify √27,
  • can you simplify √15? Factors of 15 are 1, 3, 5, 15. There are no perfect squares in those numbers, so this square root can't be simplified.

So, how to simplify square roots? To explain that, we will use handy square root property we have talked about earlier, namely, the alternative square root formula:

√x = x^(1/2)

We can use those two forms of square roots and switch between them whenever we want. Particularly, we remember that power of multiplication of two specific numbers is equivalent to the multiplication of those specific numbers raised to the same powers. Therefore, we can write that:

(x * y)^(1/2) = x^(1/2) * y^(1/2) ⟺ √(x * y) = √x * √y,

How can you use this knowledge? The argument of a square root is usually not a perfect square you can easily calculate, but it may contain a perfect square amongst its factors. In other words, you can write it as a multiplication of two numbers, where one of the numbers is the perfect square, e.g., 45 = 9 * 5 (9 is a perfect square). The requirement of having at least one factor that is a perfect square is necessary to simplify the square root. At this point, you should probably know what the next step will be. You need to put this multiplication under the square root. In our example:

√45 = 45^(1/2) = (9 * 5)^(1/2) = 9^(1/2) * 5^(1/2) = √9 * √5 = 3√5.

You have successfully simplified your first square root! Of course, you don't have to write down all these calculations. As long as you remember that square root is equivalent to the power of one half, you can shorten them. Let's practice simplifying square roots with some other examples:

  • How to simplify square root of 27? √27 = √(9 * 3) = √9 * √3 = 3√3;
  • How to simplify square root of 8? √8 = √(4 * 2) = √4 * √2 = 2√2;
  • How to simplify square root of 144? √144 = √(4 * 36) = √4 * √36 = 2 * 6 = 12.

In the last example, you didn't have to simplify square root at all, because 144 is a perfect square. You could just remember that 12 * 12 = 144. However, we wanted to show you that with the process of simplification, you can easily calculate square roots of perfect squares too. It is useful when dealing with big numbers.

Finally, you may ask how to simplify roots of higher orders, e.g., cube roots. In fact, the process is very analogical to the square roots, but in the case of cube roots, you have to find at least one factor that is a perfect cube, not a perfect square, i.e., 8 = 2³, 27 = 3³, 64 = 4³, 125 = 5³ and so on. Then you divide your number into two parts and put under the cube root. Let's take the following example of simplifying ³√192:

∛192 = ∛(64 * 3) = ∛64 * ∛3 = 4∛3

It may seem a little bit complicated at first glance, but after some practice, you will be able to simplify roots in your head. Trust us!

Adding, subtracting, multiplying and dividing square roots

Adding square roots and subtracting square roots

Unfortunately, subtracting or adding square roots are not as easy as subtracting/adding the regular numbers. For example, if 2 + 3 = 5, it doesn't mean that √2 + √3 equals √5. That's wrong! To understand why is that, imagine that you have two different types of shapes: triangles 🔺 and circles 🔵. What happens when you add one triangle to one circle 🔺 + 🔵? Nothing! You still have one triangle and one circle 🔺 + 🔵. On the other hand, what happens when you try to add three triangles to five triangles: 3🔺 + 5🔺? You'll we get eight triangles 8🔺.

Adding square roots is very similar to this. The result of adding √2 + √3 is still √2 + √3. You can't simplify it further. However, different situation is when both square roots have the same number under the root symbol. Then we can add them just as regular numbers (or triangles). For example 3√2 + 5√2 equals 8√2. The same thing is with subtracting square roots. Let's take a look on more examples illustrating this square root property:

  • What is 6√17 + 5√17? Answer: 6√17 + 5√17 = 11√17;
  • What is 4√7 - 7√7? Answer: 4√7 - 7√7 = -3√7;
  • What is 2√2 + 3√8? Answer: 2√2 + 3√8 = 2√2 + 6√2 = 8√2, because we simplified √8 = √(4 * 2) = √4 * √2 = 2√2;
  • What is √45 - √20? Answer: √45 - √20 = 3√5 - 2√5 = √5, because we simplified √45 = √(9 * 5) = √9 * √5 = 3√5 and √20 = √(4 * 5) = √4 * √5 = 2√5;
  • What is 7√13 + 2√22? Anwer: 7√13 + 2√22, we can't simplify this further;
  • What is √3 - √18? Anwer: √3 - √18 = √3 - 3√2, we can't simplify this further than this, but we at least simplified √18 = √(9 * 2) = √9 * √2 = 3√2.

Multiplying square roots and dividing square roots

Now, when adding square roots is a piece of cake for you, let's go one step further. What about multiplying square roots and dividing square roots? Don't be so scared! In fact, you already did it during the lesson of simplifying square roots. Multiplying square roots bases on the square root property that we have used before a few times, that is:

√x = x^(1/2)

Do you remember how to multiply numbers that are raised to the same power? As a reminder:

xⁿ * yⁿ = (x * y)ⁿ,

and therefore

x^(1/2) * y^(1/2) = (x * y)^(1/2) ⟺ √x * √y = √(x * y).

As opposed to addition, you can multiply every two square roots. Remember that multiplication has commutative property, that means that the order in which two numbers are multiplied does not matter. Few examples should clarify this problem:

  • What is √3 * √2? Answer: √3 * √2 = √6;
  • What is 2√5 * 5√3? Answer: 2√5 * 5√3 = 2 * 5 * √5 * √3 = 10√15, because multiplication is commutative;
  • What is 2√6 * 3√3? Answer: 2√6 * 3√3 = 2 * 3 * √6 * √3 = 6√18 = 18√3, we simplified √18 = √(9 * 2) = √9 * √2 = 3√2.

Dividing square root is almost the same since:

x^(1/2) / y^(1/2) = (x / y)^(1/2) ⟺ √x / √y = √(x / y).

All you need to do is to replace multiplication sign with a division. However, the division is not a commutative operator! You have to calculate numbers that stand before the square roots and numbers under the square roots separately. As always, some practical examples:

  • What is √15 / √3? Answer: √15 / √3 = √5;
  • What is 10√6 / 5√2? Answer: 10√6 / 5√2 = (10 / 5) * (√6 / √2) = 2√3;
  • What is 6√2 / 3√5? Answer: 6√2 / 3√5 = (6 / 3) * (√2 / √5) = 2√(2/5) = 2√(0.4), we switched there from a simple fraction 2/5 to the decimal fraction 2/5 = 4/10 = 0.4.

Square roots of exponents and fractions

Calculating the square root of the exponent or square root of the fraction might not be clear for you. But with the knowledge you acquired in the previous paragraph, it shouldn't be that hard to understand! Let's begin with the square roots of exponents. In that case, it will be easier for you to use the alternative form of square root √x = x^(1/2). Do you remember what is the power rule? If not, there is a quick reminder:

(x^n)^m = x^(n*m),

where n and m are any real numbers. Now, when you place 1/2 instead of m you'll get nothing else but a square root:

√(x^n) = (x^n)^(1/2) = x^(n/2),

and that's how you find the square root of an exponent. Speaking of exponents, above equation looks very similar to the standard normal distribution density function, which is widely used in statistics.

If you're still not sure about taking square roots of exponents, here are a few examples:

  • square root of 2^4: √(2^4) = (2^4)^(1/2) = 2^(4/2) = 2^2 = 4,
  • square root of 5^3: √(5^3) = (5^3)^(1/2) = 5^(3/2),
  • square root of 4^5: √(4^5) = (4^5)^(1/2) = 4^(5/2) = (2^2)^(5/2) = 2^5 = 32.

As you can see, sometimes it is impossible to get as pretty result as in the first example. However, in the third example, we showed you a little trick with expressing 4 as 2^2. This approach can often simplify more complicated equations.

What about square roots of fractions? Take a look at the previous section where we write about dividing square roots. You can find there the following relation that should explain everything:

(x / y)^(1/2) ⟺ √x / √y = √(x / y),

where x / y is a fraction. Below you can find some examples of square roots of a fraction:

  • square root of 4/9: √(4/9) = √4 / √9 = 2/3,
  • square root of 1/100: √(1/100) = √1 / √100 = 1/10,
  • square root of 1/5: √(1/5) = √1 / √5 = 1/√5 = √5/5.

Leaving roots in the denominator is not a very good habit. That's why we got rid of it in the last example. We just multiplied both, numerator and denominator, by the same number (we can always do that), in this case by √5.

Square root function and graph

Functions play a vital role not only in mathematics but in many other areas like physics, statistics, or finance. Function f(x) is nothing more than a formula that says how the value of f(x) changes with the argument x. To see some examples, check out our finance tools made by financial specialists, for example, the compound interest calculator or future value calculator. You will find there some functions that you can apply in real life. They're a great help if you want to know how to calculate the compound interest or to estimate the future value of an annuity.

Below you can find the square root graph, made up of half of a parabola. Check it and try to validate, for example, whether the square root function of x = 9 is 3 and of x = 16 is 4 (as it should be).

Let's go back to the square root function f(x) = √x and explore what are its basic properties. We consider there only the positive part of f(x) (as you can see in the square root graph above). So, the square root function:

  • is continuous and growing for all non-negative x,
  • is differentiable for all positive x (see derivative of the square root section for more information),
  • approaches the limit of infinity as x approaches infinity (lim √x → ∞ when x → ∞),
  • is a real number for all non-negative x and a complex number for all negative x (we write more about it in the square root of a negative number section).

You probably might already notice that the square root of an area of a square gives its length. This feature is used in one of our construction calculators - square footage calculator. If you plan to do any renovation in the future, these tools might be a great help. Don't forget to use them!

Derivative of the square root

A derivative of a function tells us how fast does this function change with its argument. One of the simplest example in physics is the position of the object and its velocity (the rate of change of position). Let's say that function x(t) describes how the distance of the moving car from the specific point changes with time t. Do you know what determines how fast does a distance you traveled increase? The answer is the speed of the car! So the derivative of the position x(t) is velocity v(t)(velocity can depend on time too). To denote derivative, we usually use apostrophe v(t) = x'(t) or the derivative symbol v(t) = dx(t)/dt.

The derivative of the general function f(x) is not always easy to calculate. However, in some circumstances, if the function takes a specific form, we've got some given formulas. For example, if

f(x) = x^n,

where n is any real number, the derivative is as follows:

f'(x) = n * x^(n-1).

It may not look like, but this answers the question what is the derivative of a square root. Do you remember the alternative (exponential) form of a square root? Let us remind it:

√x = x^(1/2).

You can see that in this case n = 1/2, so the derivative of a square root is:

(√x)' = (x^(1/2))' = 1/2 * x^(-1/2) = 1/(2√x).

Since number to the negative power is one over that number so estimation of derivation will involve fractions. We've got a tool that could be essential when adding or subtracting fractions with different denominators. It is called LCM calculator, and it tells you how to find the Least Common Multiple.

A derivative of a square root is needed to obtain the coefficients in the so-called Taylor expansion. We don't want to go into details too deeply. So briefly, Taylor series allows you to approximate various functions with the polynomials that are much easier to calculate. For example, the Taylor expansion of √(1 + x) about the point x = 0 is given by:

√(1 + x) = 1 + 1/2 * x - 1/8 * x² + 1/16 * x³ - 5/128 * x⁴ + ...,

which is valid for -1 ≤ x ≤ 1. Although the above expression has an infinite number of terms, for the approximate value you can use just a few first terms. Let's try it! With x = 0.5 and first five terms, you get:

√(1.5) = 1 + 1/2 * 0.5 - 1/8 * 0.25 + 1/16 * 0.125 - 5/128 * 0.0625,

√(1.5) ≈ 1.2241,

and the real value, provided by our calculator, is √(1.5) ≈ 1.2247. Close enough!

That was a lot of maths and equations so far. For those of you who are persistent enough, we've prepared the next section which explains how to calculate the square root of a negative number.

Square root of a negative number

At school, you probably have been taught that square root of a negative number does not exist. This is true when you consider only real numbers. Long time ago, to perform advanced calculations, mathematicians had to introduce a more general set of numbers - the complex numbers. They can be expressed in the following form:

x = a + b*i,

where x is the complex number with the real part a and imaginary part b. What differs complex number from real ones is the imaginary number i. Here you have some examples of complex numbers: 2 + 3i, 5i, 1.5 + 4i, 2. You may be surprised seeing 2 there which is a real number. Yes, it is, but it is also a complex number with b = 0. Complex numbers are a generalization of the real numbers.

So far imaginary number i is probably still a mystery for you. What is it at all? Well, although it may look weird, it is defined by the following equation:

i = √(-1),

and that's all what you need to calculate square root of every number, whether it is positive or not. Let's see some examples:

  • square root of -9: √(-9) = √(-1 * 9) = √(-1)√9 = 3i,
  • square root of -13: √(-13) = √(-1 * 13) = √(-1)√13 = i√13,
  • square root of -49: √(-49) = √(-1 * 49) = √(-1)√49 = 7i.

Isn't that simple? This problem doesn't arise with the cube root since you can obtain the negative number by multiplying three of the identical negative numbers (which you can't do with two negative numbers). For example:

∛(-64) = ∛[(-4)*(-4)*(-4)] = -4.

That's probably everything you should know about square roots. We appreciate you stayed with us until this point! As a reward you should bake something sweet for yourself :-) To get the appropriate amount of ingredients, you can use our grams to cups calculator. It can work in both ways, i.e., to convert grams to cups and convert grams to cups. And if you ask yourself "How many calories should I eat a day?", visit our handy calorie calculator!

Mateusz Mucha, Piotr Małek and Dominik Czernia, PhD student

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