Laws of Exponents and Radicals
Laws of Exponents (Index Law)
1. $x^n = x \cdot x \cdot x ... \, (n \text{ factors})$
2. $x^m \cdot x^n = x^{m + n}$
3. $(x^m)^n = x^{mn}$
4. $(xyz)^n = x^n \, y^n \, z^n$
5. $\dfrac{x^m}{x^n} = x^{m - n}$
6. $\left( \dfrac{x}{y} \right)^n = \dfrac{x^n}{y^n}$
7. $x^{-n} = \dfrac{1}{x^n}$ and $\dfrac{1}{x^{-n}} = x^n$
8. $x^0 = 1$, provided $x \ne 0$.
9. $(x^m)^{1/n} = (x^{1/n})^m = x^{m/n}$
10. $x^{m/n} = \sqrt[n]{x^m}$
11. If $x^m = x^n$, then $m = n$ provided $x \ne 0$.
Properties of Radicals
1. $\sqrt[n]{x} = x^{1/n}$
2. $\sqrt[n]{x^m} = \left( \sqrt[n]{x} \right)^m = x^{m/n}$
3. $\sqrt[n]{x} \, \sqrt[n]{y} = \sqrt[n]{xy}$
4. $\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}}$
5. $\sqrt[n]{x} \, \sqrt[m]{x} = \sqrt[mn]{x^{m + n}}$
6. $\dfrac{\sqrt[n]{x}}{\sqrt[m]{x}} = \sqrt[mn]{x^{m - n}}$
7. $\left( \sqrt[n]{x} \right)^n = x$