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.

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.

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However, every positive number has two square roots, the positive sqrt also known as principal square root \sqrt{a}, and the negative sqrt -\sqrt{a}.

Together, this is written as \pm\sqrt{a} or \pm\sqrt[2]{a}, 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 \sqrt{16} = ±4.

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.
Square Root

How to Find the Square Root of a Number

The easiest way to find the square root of a number is using a calculator like the one you can find in the next section a little bit further down.

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

Square Root Calculator

Our calculator is straightforward, and it can calculate the root of any non-negative real number. Just enter a valid number; you then automatically obtain both, the principal as well as the negative result. To start over, press reset first.

Calculate \sqrt[2]{x}

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

Square Root Property

With a,b \in \mathbb{R^{+}} and k,m \in \mathbb{N}, the properties are as follows:

  • \sqrt{a^{2}} = |a| = -a if a < 0 and a if ≥ 0
  • \sqrt{ab}= \sqrt{a} \sqrt{b}
  • \sqrt{a^{m}} = a^{\frac{m}{2}} \Rightarrow \sqrt{a} = a^{\frac{1}{2}}
  • \sqrt{a/b}= \frac{\sqrt{a}}{\sqrt{b}}
  • \sqrt[2]{a} = \sqrt{a}
  • \sqrt{0} = 0
  • \sqrt{a^{-m}} = \frac{1}{\sqrt{a^{m}}}
  • \sqrt{a^{m}} = \sqrt[k]{a^{km}}
  • \sqrt{\sqrt[m]{a}} = \sqrt[2m]{a} = \sqrt[m]{\sqrt{a}}

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:

  • \sqrt{9} = \pm 3
  • \sqrt{400} = \sqrt{16} \sqrt{25} = 4 x 5 = 20
  • \sqrt{4^{4}} = 2^{\frac{4}{2}} = 2^{2} = 4
  • \sqrt{9/4}= \frac{\sqrt{9}}{\sqrt{4}} = 3/2
  • \sqrt{3^{-2}} = \frac{1}{\sqrt{3^{2}}} = \frac{1}{\sqrt{9}} = 1/3
  • \sqrt[k]{6^{k4}} = \sqrt{6^{4}} = \sqrt{1296} = 36
  • \sqrt{\sqrt[3]{125}} = \sqrt[2x3]{125}
  • = \sqrt[3]{\sqrt{125}} = 2.23607...

Frequently searched terms on this site include:

Square Root of Negative Number

In \mathbb{R} negative numbers don’t have a square root because the square of any real number will be 0 or positive. For example you wont find any square number -16.

But for imaginary numbers this does exist in the form \sqrt{-16}=4i

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

\sqrt{-x}=\pm i\sqrt{x}\hspace{15px} x < 0, x \in \mathbb{R}

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

Square Root Function

Last but not least, here is the square root function f(x) = √x, x \in \mathbb{R}, x ≥ 0
Square Root Function

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

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.

Table of Squares and Square Roots

NumberSquareSquare Root
000
111
241.4142135624
391.7320508076
4162
5252.2360679775
6362.4494897428
7492.6457513111
8642.8284271247
9813
101003.1622776602
111213.3166247904
121443.4641016151
131693.6055512755
141963.7416573868
152253.8729833462
162564
172894.1231056256
183244.2426406871
193614.3588989435
204004.472135955
214414.582575695
224844.6904157598
235294.7958315233
245764.8989794856
256255
266765.0990195136
277295.1961524227
287845.2915026221
298415.3851648071
309005.4772255751
319615.5677643628
3210245.6568542495
3310895.7445626465
3411565.8309518948
3512255.9160797831
3612966
3713696.0827625303
3814446.164414003
3915216.2449979984
4016006.3245553203
4116816.4031242374
4217646.4807406984
4318496.5574385243
4419366.6332495807
4520256.7082039325
4621166.7823299831
4722096.8556546004
4823046.9282032303
4924017
5025007.0710678119
5126017.1414284285
5227047.2111025509
5328097.2801098893
5429167.3484692283
5530257.4161984871
5631367.4833147735
5732497.5498344353
5833647.6157731059
5934817.6811457479
6036007.7459666924
6137217.8102496759
6238447.874007874
6339697.9372539332
6440968
6542258.0622577483
6643568.1240384046
6744898.1853527719
6846248.2462112512
6947618.3066238629
7049008.3666002653
7150418.4261497732
7251848.4852813742
7353298.5440037453
7454768.602325267
7556258.6602540378
7657768.7177978871
7759298.7749643874
7860848.8317608663
7962418.8881944173
8064008.94427191
8165619
8267249.0553851381
8368899.1104335791
8470569.1651513899
8572259.2195444573
8673969.2736184955
8775699.3273790531
8877449.3808315196
8979219.4339811321
9081009.4868329805
9182819.5393920142
9284649.5916630466
9386499.643650761
9488369.6953597148
9590259.7467943448
9692169.7979589711
9794099.8488578018
9896049.8994949366
9998019.9498743711
1001000010

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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