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 = 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.

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 $\pm\sqrt{a}\hspace{3px}$ 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 square root 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 of a square root 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 square root 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 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:

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.

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.

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.

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.

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.

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 square root 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.

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

Square Root Property

With $a,b\hspace{3px} \varepsilon\hspace{3px} \mathbb{R^{+}}$ and $k,m \hspace{3px} \varepsilon\hspace{3px} \mathbb{N}$, the properties of square roots are as follows:

• $\sqrt{a^{2}} = |a|\hspace{3px}$ = -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 square root 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 square root examples in the order of appearance:

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

Frequently searched square roots on this site include:

• Square Root of 123
• Square Root of 289
• Square Root of 400

Square Root of Negative Number

In $\mathbb{R}\hspace{3px}$ 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 $\epsilon\hspace{3px}$ $\mathbb{R}$

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, $x \varepsilon \mathbb{R}$, 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.

This brings us to the end of our article. Note that you can find many square roots 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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More Information:

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