Square Root

A square root of $ x $ (or radical of $ x $) is a mathematical concept noted $ \sqrt{x} $ (ou sqrt(x)) that refers to the number that, when multiplied by itself, produces the number $ x $.

There are several methods to calculate a root square.

— By hand framing: the classic method is to estimate the value by calculating which integers squared would give a minimum interval.

— By extraction of squares: if the number under the root is factorized with squares, then it is possible to extract them from the root.

— With a square root calculator like this one from dCode:

Enter a positive or negative number (in this case, it will have complex roots).

Choose the format of the result, either an exact value (if it is an integer or variables) or approximate (decimal number with adjustable precision by defining a minimum number of significant digits)

For any real number $ a \in \mathbb{R} $

$$ \sqrt{a^2} = |a| $$

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

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

For any number $ b $

$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \\ \sqrt{ \frac{a}{b} } = \frac{\sqrt{a}}{\sqrt{b}} \qquad (b \neq 0) \\ \sqrt{a^2 \times b} = |a| \sqrt{b} $$

The square root of a perfect square is an integer.

The simplification of a square root generally passes by the factorization of the component under the root by one or more squares.

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.

Exponentiation by the value $ 1/2 $ is also common: $ \sqrt{x} = x^{1/2} $ (exponent 0.5)

Terms root, radix ou radicand sont équivalents.

Square roots are needed in many areas of mathematics.

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

A square number is the square of an integer.

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

The square root of a negative number is not a real number. It belongs to the complex numbers and is written in the form $ i \sqrt{|a|} $, where $ i $ is the imaginary unit, defined by $ i^2 = -1 $.

The square root of zero is zero, because $ 0 \times 0 = 0 $

The square root of one is one, because $ 1 \times 1 = 1 $