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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   $x \ne 0$.