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Rooting


Concept of Rooting

Rooting or the root operation is a mathematical operation used to find the base of a given power using the exponent and base:

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


Properties of Roots

  • Every positive number a has one n-th root.
  • Negative numbers do not have an even root.
  • Negative numbers have one odd root, which is also negative.

\begin{align} \sqrt[2n]{a^{2n}}&=\left | a \right |\\ \\ \sqrt[2n+1]{a^{2n+1}}&=a\\ \\ \sqrt[n]{0}&=0,\; \textrm{where}\; n\neq 0\\ \\ \sqrt[n]{1}&=1,\; \textrm{where}\; n\neq 0\\ \end{align}


Operations with Roots

Taking into account the properties of roots mentioned above:

\begin{align} \sqrt[n]{a \times b}&=\sqrt[n]{a} \times \sqrt[n]{b}\\ \\ \sqrt[n]{\frac {a}{b}}&=\frac {\sqrt[n]{a}}{\sqrt[n]{b}}\\ \\ \sqrt[m]{\sqrt[n]{a}}&=\sqrt[m \times n]{a}\\ \\ \sqrt[n]{a^{m}} &= a^{\frac{m}{n}}\\ \\ (\sqrt[n]{a})^{m} &= \sqrt[n]{a^{m}}\\ \\ \sqrt[n]{a^{m}} &= \sqrt[k \times n]{a^{k \times m}}\\ \end{align}


See also: