Exponents - Powers and Roots
The laws of fractional and integer exponents including formulas for integer and fractional exponent operations.
Integer Exponents
Law 1: $a^n \cdot a^m = a^{n+m}$
Law 2: $\frac{a^n}{a^m} = a^{n-m}$
Law 3: $(a^n)^m = (a^m)^n = a^{nm}$
Law 4: $(ab)^m = a^m b^m$
Law 5: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$
Law 6: $p a^n \pm q a^n = (p \pm q) a^n$
Law 7: $a^{-n} = \frac{1}{a^n}$
Fractional Exponents
Law 8: $a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m$
Law 9: $p a^{1/n} \pm q a^{1/n} = (p \pm q) a^{1/n}$
Law 10: $(ab)^{1/n} = a^{1/n} b^{1/n}$
Law 11: $\frac{a^{1/n}}{b^{1/n}} = \left(\frac{a}{b}\right)^{1/n}$
Law 12: $(a^{mx})^{1/nx} = (a^m)^{1/n}$
Law 13: $(-a)^{1/2} = i a^{1/2}$ (where $i$ is the imaginary number)