Square Root

In math, the square root b of a number a is such that b² = a. By definition, when you multiply b by itself two times you get the value of a.

A square root is usually denoted √a, but it can also be written in exponential form with the base a and the exponent 1/2 as explained further below in this article.

Read on to learn everything about these numbers, including the properties, and make sure to check out our calculator.

Calculator

Number:

Square Root:

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If you happen to know exponentiation then you can think of the square root (sqrt) of a number as the inverse operation to elevating a number to the power of two.

Whereas in exponentiation elevating a number a to the power of two is defined as a² = b, the sqrt b is defined as b = a^1/2.

For example with a = 16 we get:

√16 = (4²)^1/2 = 4^2/2 = 4¹ = 4.

In other words, the sqrt of 16 is 4, because 4 times 4 is 16.

In the same fashion, the square root of 9 = 3.

However, every positive number has two square roots, the positive sqrt also known as principal square root

and the negative sqrt


Together, this is written as

or

but the index of 2 is usually omitted.
In the example above where a is 16, the number -4 is the negative root because (-4)² equals 16, too.

So we can say

Square Root Symbol

The symbol √ is called radix, or more commonly, radical sign.

In Microsoft Word for example, the sqrt symbol can be found in the Insert menu, Symbol, Mathematical Operators.

Alternatively, you can use the numeric keyboard by pressing ALT + 251.

The √ symbol, resembling the lowercase letter “r” to indicate radix, was introduced by the Austrian mathematician Christoph Rudolff in his book Coss which was first published in 1525.

Square Root Parts

The parts are as follows:

The radix sign tell us that it is a mathematical root, and the index of two tells us that it is the second root.

The number below the radix is the radicand.

The result of the mathematical operation is denoted by the equal sign and called the root.

How to Find the Square Root of a Number

The easiest way to produce the square root of a number is using a calculator like the one you can see in the first section.

In the absence of a calculator you can use the guess and check method:

1. Find the two perfect squares your number is between. The sqrt of your number must be between the roots of these perfect squares. For example, to produce √22 proceed as follows: 22 lies between the perfect square of 4, 16, and the perfect square of 5, 25. Therefore, √22 must be between √16 and √25, that is between 4 and 5.
2. Build the sum of these two roots to obtain 9, and divide the result by 2 to get 4.5. Then raise it to the power of 2: (4.5)² = 20.25. The result is less than 22, so √22 must be bigger than 4.5.
3. Build the sum of 4.5 and 5, divide it by 2 and ^2 4.75 to obtain 22.5625. This is more than 22, so √22 must be less than 4.75.
4. Next, build the sum of 4.5 and 4.75 and divide it by 2. Then elevate 4.625 to the power of 2 to obtain 21.390625, less than 22. Thus, √22 must be bigger than 4.625.
5. Sum 4.625 + 4.75, and square half of it: (4.6875)² = 21.97265625. This is very close to 22, so √22 is just a little bit bigger.
6. If you need more precision proceed as above until your result is close enough by summing and dividing the result by 2, then square it.

We hope this answered the question how to find square root manually.

Practice

• Find the value of √121.
• Solve √14 = ?
• What is the √ symbol called?

If necessary, you may look up the solution in the chart further down.

For methods of computing these numbers please check the reference section at the end of this page.

About our Square Root Calculator

Our calculator at the beginning of this article is straightforward, and it can calculate the root of any real number.

Just enter a valid input; you then automatically obtain both, the principal as well as the negative result.

To start over, press reset first.

Observe that our tool works both ways; that is the math is bidirectional.

Use the up and down arrows (known as spinner) to increase or decrease the input value.

You may change both, the upper as well as the lower input field.

On this website our calculators usually only compute real numbers, but our square root calculator is different:

It does compute complex values for negative input numbers such as -1, -2, -3, etc.

If our tool has been useful to you bookmark it now as square root calculator.

Square Root Property

With and , the properties are as follows:

• = -a if a < 0 and a if ≥ 0

The most important property is the first one as the negative number tends to be forgotten.

Read on to see the properties in use:

Square Root Examples

We use the list of properties above to show you some examples in the order of appearance:

Frequently searched terms on this site include:

• Square Root of 123
• Square Root of 194
• Square Root of 400

Practice 2

a) What is √40 / √10 ?
b) Simplify √3 x √12 = ?
c) Solve √? = √16 x √9

Answers:

a) √40 / √10 = √40/10 = √4 = ±2.

b) √3 x √12 = √3×12 = √36 = ±6.

c) √144 = √16 x √9 = ±(4 x 3) ±12.

Square Root of Negative Number

In the set of real numbers negative numbers don’t have a square root because the square of any real number will be 0 or positive (non-negative).

For example you wont come across any square number -16.

But for imaginary numbers this does exist in the form

So, the sqrt of any negative number is imaginary and as follows:

The most famous negative square root is that of the number –1 which you can locate here square root of negative 1.

As stated before, our app can compute the square root of a negative number.

Give it a try now!

This table is a shortcut to the most searched items in Google:

Square Root Function

Last but not least, here is the square root function f(x) = √x, , x ≥ 0



This function maps the set real numbers equal or greater than zero onto the principal root. In geometry, the function f(x) = √x maps the area of a square to its side length.

If something remains unclear do not hesitate getting in touch with us.

We are constantly trying to improve our site, and truly appreciate your feedback.

By the way: Websites which are related to this one and you may also be interested in can be found in the “recommended sites” section of the sidebar.

Table of Squares and Square Roots

Bottom Line

This brings us to the end of our article. Note that you can find many numbers by using the search form in the sidebar.

There, you can also search for cube roots, squares, cubes, perfect squares as well as perfect cubes.

If you are happy with our content then please press the social buttons below to let your friends know about our website.

For questions and comments use the form below.

We try to answer all questions quickly, yet, for time constraints we currently don’t provide tuition.

But we really like to hear from you, in particular about our new calculator and the accordion with the collapsible content.

Nota Bene: A term closely related to square roots is “perfect square”.

You can learn everything about perfect squares on our article Squared Numbers, located in the header menu of this page.

Last, but not least, make sure to install our absolutely free free PWA app (see menu or sidebar), or to bookmark us.

And come back soon!

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More Information:

• Square root on Wikipedia
• Methods of computing square roots on Wikipedia

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