Square Root Calculator and Simplifier - SQRT Online (2024)

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 $.

Example: The square root of $ 9 $ is $ 3 $ that is written $ \sqrt{9} = 3 $, because $ 3 \times 3 = 9 $

Generally, numbers have 2 roots, a positive and a negative, but the negative is usually omitted.

Example: It would be more accurate to write: the square roots of $ 9 $ are $ 3 $ and $ -3 $ which is written $ \sqrt{9} = \pm 3 $, indeed, $ 3 \times 3 = 9 = (-3) \times (-3) = 9 $

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.

Example: Enclosing $ \sqrt{8} $: $ 2^2 = 4 < 8 < 9 = 3^3 $ so $ 2 < \sqrt{8} < 3 $, it is then possible to enclose the first digit after the comma: $ 2.8^2 < 8 < 2.9^2 $ etc.

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

Example: Factorization of $ \sqrt{8} = \sqrt{ 4 \times 2 } = \sqrt{ 2^2 \times 2 } = 2 \sqrt{2} $. Since $ \sqrt{2} \approx 1.414 $, then $ \sqrt{8} \approx 2.828 $

— 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)

Example: $ \sqrt{12} = 2 \sqrt{3} \approx 3.464 $

Example: $ \sqrt{-1} = i $ (complex root)

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

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

For any number $ b $

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

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

Example: $ \sqrt{20} = \sqrt{ 2^2 \times 5 } = \sqrt{ 2^2 } \times \sqrt{ 5 } = 2 \sqrt{ 5 } $

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.

Square roots are needed in many areas of mathematics.

Example: In algebra: in algebraic calculations, the roots are used to solve polynomial equations of the type $ x^2 + 2x + 1 = 0 $

Example: In geometry: in length calculations (or vector norms), roots are used to find solutions to the Pythagorean theorem $ a^2 + b^2 = c^2 $

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.