Radicals

For positive integers $ m,n$ and real numbers $ a,b$ such that $ \sqrt[n]{a}$ and $ \sqrt[n]{b}$ are real:

  $\displaystyle \sqrt{a}=a^{1/2}$   $\displaystyle \sqrt[n]{a}=a^{1/n}$   $\displaystyle a^{m/n}=(a^{1/n})^m=(a^m)^{1/n}=\sqrt[n]{a^m}$      
  $\displaystyle \sqrt[n]{ab}=\Big(\sqrt[n]{a}\Big)\Big(\sqrt[n]{b}\Big)$   $\displaystyle \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},b\ne 0$   $\displaystyle \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$      

$ \quad\quad\quad\quad (\sqrt[n]{a})^n=a
\quad\quad\quad\quad\quad\quad\quad \s...
...ght\rvert , & \text{for $n$ even} \\
a, & \text{for $n$ odd} \\
\end{cases}$




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