# nth root formulas

In the following formulas, $a$ is a nonnegative real number and other letters positive integers.  For other formulas, see the parent entry (http://planetmath.org/NthRoot).

1. 1.

$\sqrt[n]{0}\;=\;0$,   $\sqrt[n]{1}\;=\;1$

2. 2.

$\sqrt[1]{a}\;=\;a$

3. 3.

$\sqrt[m]{\sqrt[n]{a}}\;=\;\sqrt[mn]{a}\;=\;\sqrt[n]{\sqrt[m]{a}}$

4. 4.

$\sqrt[nk]{a^{mk}}\;=\;\sqrt[n]{a^{m}}$

5. 5.

$\sqrt[n]{a^{m}}\;=\;(\sqrt[n]{a})^{m}$

6. 6.

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

Proof.  For proving, one uses the definition of http://planetmath.org/node/754$n$th root and the power (http://planetmath.org/GeneralAssociativity) laws.

$1^{\circ}$.  $0^{n}=0,\quad 1^{n}=1$

$2^{\circ}$.  $a^{1}=a$

$3^{\circ}$.  $(\sqrt[m]{\sqrt[n]{a}})^{mn}\;=\;((\sqrt[m]{\sqrt[n]{a}})^{m})^{n}\;=\;(\sqrt[% n]{a})^{n}\;=\;a$

$4^{\circ}$.  $(\sqrt[n]{a^{m}})^{nk}\;=\;((\sqrt[n]{a^{m}})^{n})^{k}\;=\;(a^{m})^{k}\;=\;a^{mk}$

$5^{\circ}$.  $((\sqrt[n]{a})^{m})^{n}\;=\;((\sqrt[n]{a})^{n})^{m}\;=\;a^{m}$

$6^{\circ}$.  $(\sqrt[m]{a}\cdot\sqrt[n]{a})^{mn}\;=\;(\sqrt[m]{a})^{mn}(\sqrt[n]{a})^{mn}\;=% \;((\sqrt[m]{a})^{m})^{n}((\sqrt[n]{a})^{n})^{m}\;=\;a^{n}a^{m}\;=\;a^{m+n}$

Title nth root formulas NthRootFormulas 2014-10-25 17:59:04 2014-10-25 17:59:04 pahio (2872) pahio (2872) 12 pahio (2872) Theorem msc 26A09 root formulas root formulae FractionPower