Exponents and Radicals

For real numbers $a, b$ and integers $m, n$:

1. Product Rule: $a^m \cdot a^n = a^{m+n}$
2. Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
3. Power Rule: $(a^m)^n = a^{mn}$
4. Power of a Product: $(ab)^n = a^n b^n$
5. Power of a Quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
6. Zero Exponent: $a^0 = 1$ (for $a \neq 0$)
7. Negative Exponent: $a^{-n} = \frac{1}{a^n}$

Exponents can be fractions. The general rule is: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ Specifically: $a^{1/2} = \sqrt{a}, \quad a^{1/3} = \sqrt[3]{a}$