Square Root

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.