nth Root

Assumption $0\le a\in \mathbb{R},n\in \mathbb{N}$

$x^n=a⟺x=\sqrt[n]{a}$

• $\sqrt[n]{a^m}=(a^m{)}^{1/n}=a^{m/n}=(a^{1/n}{)}^m=(\sqrt[n]{a}{)}^m$

• $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$

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

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

• if $n$ is odd, then $\sqrt[n]{a^n}=a$

• if $n$ is even, then $\sqrt[n]{a^n}=|a|$

• The conjugate of an expression of the form $a+b\sqrt{d}$ is $a-b\sqrt{d}$, (provided that $\sqrt{d}$ does not appear in $a$ and $b$).

  • One says also that the two expressions are conjugate.