Rules for Exponents and Radicals

  • $\displaystyle{x^n = \underbrace{x \cdot x \cdot x \cdots x}_{n\textrm{ occurances of } x}}$ for integers $n \ge 1$

  • $\displaystyle{x^0 = 1 \quad \textrm{for any } x \neq 0}$

  • $\displaystyle{x^{-n} = \frac{1}{x^n}, \quad \textrm{for any } x \ne 0}$

  • $x^{1/n}$ for integers $n \ge 2$ is a value that when raised to the $n^{\textrm{th}}$ power equals $n$.
    When $n$ is even and $x \ge 0$, $x^{1/n} \ge 0$.
    When $n$ is even and $x \lt 0$, $x^{1/n}$ is undefined.

  • $\displaystyle{\sqrt[n]{x} = x^{1/n} \quad \textrm{for integers } n \ge 2}$

  • $\displaystyle{x^{m/n} = \left( x^{1/n} \right)^m = \left( \sqrt[n]{x} \right)^m \quad \textrm{ provided that } x^{1/n} \textrm{ is a real number}}$

  • $\displaystyle{x^m \cdot x^n = x^{m+n}}$

  • $\displaystyle{(x^m)^n = x^{m\,n}}$

  • $\displaystyle{\frac{x^m}{x^n} = x^{m-n}, \quad \textrm{for any } x \ne 0}$

  • $\displaystyle{(xy)^n = x^n y^n}$

  • $\displaystyle{\left( \frac{x}{y} \right)^n = \frac{x^n}{y^n}}$

  • $\displaystyle{\begin{array}{l}
    \textrm{If } n \textrm{ is even, then } \sqrt[n]{x^n} = \left| x \right|\\
    \textrm{If } n \textrm{ is odd, then } \sqrt[n]{x^n} = x
    \end{array}}$

  • $\displaystyle{\sqrt[n]{xy^{\phantom{1}}} = \sqrt[n]{x^{\phantom{1}}} \sqrt[n]{y^{\phantom{1}}}}$

  • $\displaystyle{\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}}$

  • $\displaystyle{\sqrt[m]{\sqrt[n]{x}} = \sqrt[m n]{x}}$