Tool to compute and simplify a square root. The square root for a number N, is the number noted sqrt(N) that, multiplied by itself, equals N.

Square Root - dCode

Tag(s) : Symbolic Computation, Functions

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The dCode **square root** calculator allows both positive or negative numbers (thus having complex roots) and returns answers with an exact value or an approximate value (the precision can be adjusted by defining a minimum number of significant digits)

__Example:__ $ \sqrt{4} = 2 $ and $ \sqrt{-1} = i $

Root calculations have properties similar to exponentiation:

$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \\ \sqrt{ \frac{a}{b} } = \frac{\sqrt{a}}{\sqrt{b}} $$

For any positive real number $ a \in \mathbb{R}_+^* $

$$ \sqrt{a^2} = a \\ \left( \sqrt{a} \right)^2 = a $$

Therefore

$$ \sqrt{a^2 \times b} = a \sqrt{b} $$

If the denominator is a radical, then multiply the numerator and the denominator by it to make it disappear.

$$\frac{a}{\sqrt{b}} = \frac{a\sqrt{b}}{\sqrt{b}^2} = \frac{a\sqrt{b}}{b} $$

If the denominator is an addition or subtraction of roots, then apply the remarkable identity: $ (a+b)(a-b) = a^2-b^2 $

$$ \frac{a}{\sqrt{b}+\sqrt{c}} = \frac{a(\sqrt{b}-\sqrt{c})}{(\sqrt{b}+\sqrt{c})(\sqrt{b}-\sqrt{c})} = \frac{a\sqrt{b}-a\sqrt{c}}{b-c} $$

$$ \frac{a}{\sqrt{b}-\sqrt{c}} = \frac{a(\sqrt{b}+\sqrt{c})}{(\sqrt{b}-\sqrt{c})(\sqrt{b}+\sqrt{c})} = \frac{a\sqrt{b}+a\sqrt{c}}{b-c} $$

In Unicode format there is the character `√` (U+221A).

In computer formulas, `sqrt()` function is most often used.

Terms *root*, *radix* ou *radicand* sont équivalents.

The word `sqrt` is generally used in the formula to indicate a **square root**, the word comes from the contraction of ` square root`.

__Example:__ `sqrt(2)` = $ \sqrt{2} $

A square number is the square of an integer.

__Example:__ $ 3 $ is an integer, $ 3^2 = 3 \times 3 = 9 $ then $ 9 $ is a square number.

If the **square root** of a number $ x $ is an integer, then $ x $ is a square number.

dCode retains ownership of the online "Square Root" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Square Root" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Square Root" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, copy-paste, or API access for "Square Root" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

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square,root,sqrt,radicand,radix

Source : https://www.dcode.fr/square-root

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