# Root calculator for square roots, cubic roots, ...

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## Names of the root calculation

$\sqrt[n]{a}$

Say: nth root of a. Where a is the radical and n is the root exponent.

$\sqrt[2]{a}=\sqrt{a}$

Square root of a

$\sqrt[3]{a}$

Cubic root from a

## Root as power

$\sqrt[n]{a}={a}^{\frac{1}{n}}$

The root can also be written as a power with the exponent 1/n.

${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}={\left(\sqrt[n]{a}\right)}^{m}$

$\sqrt{a}={a}^{\frac{1}{2}}$

$\sqrt[3]{a}={a}^{\frac{1}{3}}$

$\frac{1}{\sqrt{a}}=\frac{1}{{a}^{\frac{1}{2}}}={a}^{-\frac{1}{2}}$

## Calculation rules of root calculation

$\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$

Product rule with the same root exponent

$\sqrt[m]{a}\cdot \sqrt[n]{a}=\sqrt[n\cdot m]{{a}^{n+m}}$

Product rule for the same Radikand

$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

Quotient rule

$\sqrt[m]{\sqrt[n]{a}}=\sqrt[m\cdot n]{a}$

Iteration rule

## Root equation

$\sqrt[n]{x}=a⇔x={a}^{n}$

The root equation can be transformed by means of the power laws.

$\sqrt{x}=a⇔x={a}^{2}$

For the square root, the transformation is done by squaring.

## Representation with e-function and logarithm

$\sqrt[n]{x}={e}^{\frac{\mathrm{ln}x}{n}}$

The root function can be represented by the exponential function and the logarithm.