nth Root

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

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

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

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

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

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

• Let $a\in \mathbb{R}$ and $n\in \mathbb{N}$. Then, $x^n=a⟺\{\begin{array}{ll}x=\pm \sqrt[n]{a}& \text{if }n\text{ is even and }a\ge 0\\ x=\sqrt[n]{a}& \text{if }n\text{ is odd}\\ \text{no real solution}& \text{if }n\text{ is even and }a<0\end{array}$

• 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.

Square Root

• Let $x\in \mathbb{R}$ with $x\ge 0$. Then $\sqrt{x}\in {\mathbb{R}}_{\ge 0}$ and $(\sqrt{x}{)}^2=x$.
• $\forall x\in \mathbb{R},\sqrt{x^2}=|x|$
• Let $a\in \mathbb{R}$. Then, $x^2=a⟺\{\begin{array}{ll}x=\pm \sqrt{a}& \text{if }a>0\\ x=0& \text{if }a=0\\ \text{no real solution}& \text{if }a<0\end{array}$