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.

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On this Site You Can Learn all About Square Roots! Click To Tweet 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 = a1/2.

For example with a = 16 we get:

√16 = (42)1/2 = 42/2 = 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.

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:

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.

Frequently Asked Questions

Click on the question which is of interest to you to see the collapsible content answer.

What Does √ Mean?

The √ symbol denotes a square root in mathematics, and is called radical sign, radical symbol or root symbol.

Why is Square Root Used?

In mathematics, a square root x solves an equation of the type x² = y.

How Do You Figure out the Square Root of a Number?

Start with an initial guess such that two times that value equals approximately the number, then keep improving the guess until you have the required precision. In the same way you can employ the fast-converging Newton–Raphson method.

Where Do You Use Square Roots in Real Life?

Square roots are common in algebra, geometry as well as physics; in particular in quadratic equations.

What is Square and Square Root?

The square is the concept of a number x such that x² = y, whereas the square root is the concept of a number y such that √y = x.

What Does Square Mean?

In mathematics, a square is the concept of a number y such that x² = y.

What Does Root Mean?

In mathematics, a square root is the concept of a number x such that √y = x.

What Are the Two Square Roots of 100?

The two square roots of 10 = ±10.

Does a Negative Has a Square Root?

A negative only has a square root in the set of complex numbers.

Is the Square Root of 144 a Rational Number?

√144 = ±12, which is a whole number, not a rational.

What is the Difference Between a Square and Square Root?

Square and square root are inverse functions.

Is the Square Root Always Positive?

Every real number except 0 has two real square roots, one of which is negative.
If something remains unclear do not hesitate getting in touch with us.

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Table of Squares and Square Roots

NumberSquareSquare Root
000
11±1
24±1.4142135624
39±1.7320508076
416±2
525±2.2360679775
636±2.4494897428
749±2.6457513111
864±2.8284271247
981±3
10100±3.1622776602
11121±3.3166247904
12144±3.4641016151
13169±3.6055512755
14196±3.7416573868
15225±3.8729833462
16256±4
17289±4.1231056256
18324±4.2426406871
19361±4.3588989435
20400±4.472135955
21441±4.582575695
22484±4.6904157598
23529±4.7958315233
24576±4.8989794856
25625±5
26676±5.0990195136
27729±5.1961524227
28784±5.2915026221
29841±5.3851648071
30900±5.4772255751
31961±5.5677643628
321024±5.6568542495
331089±5.7445626465
341156±5.8309518948
351225±5.9160797831
361296±6
371369±6.0827625303
381444±6.164414003
391521±6.2449979984
401600±6.3245553203
411681±6.4031242374
421764±6.4807406984
431849±6.5574385243
441936±6.6332495807
452025±6.7082039325
462116±6.7823299831
472209±6.8556546004
482304±6.9282032303
492401±7
502500±7.0710678119
512601±7.1414284285
522704±7.2111025509
532809±7.2801098893
542916±7.3484692283
553025±7.4161984871
563136±7.4833147735
573249±7.5498344353
583364±7.6157731059
593481±7.6811457479
603600±7.7459666924
613721±7.8102496759
623844±7.874007874
633969±7.9372539332
644096±8
654225±8.0622577483
664356±8.1240384046
674489±8.1853527719
684624±8.2462112512
694761±8.3066238629
704900±8.3666002653
715041±8.4261497732
725184±8.4852813742
735329±8.5440037453
745476±8.602325267
755625±8.6602540378
765776±8.7177978871
775929±8.7749643874
786084±8.8317608663
796241±8.8881944173
806400±8.94427191
816561±9
826724±9.0553851381
836889±9.1104335791
847056±9.1651513899
857225±9.2195444573
867396±9.2736184955
877569±9.3273790531
887744±9.3808315196
897921±9.4339811321
908100±9.4868329805
918281±9.5393920142
928464±9.5916630466
938649±9.643650761
948836±9.6953597148
959025±9.7467943448
969216±9.7979589711
979409±9.8488578018
989604±9.8994949366
999801±9.9498743711
10010000±10

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.

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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.

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