1. The laws of exponents are rules that describe how to simplify expressions involving powers of the same base. 2. **Product rule:** When multiplying powers with the same base, add the exponents. $$a^m \times a^n = a^{m+n}$$ 3. **Quotient rule:** When dividing powers with the same base, subtract the exponents. $$\frac{a^m}{a^n} = a^{m-n}$$ 4. **Power rule:** When raising a power to another power, multiply the exponents. $$(a^m)^n = a^{m \times n}$$ 5. **Zero exponent rule:** Any nonzero base raised to the zero power equals 1. $$a^0 = 1, \text{ for } a \neq 0$$ 6. **Negative exponent rule:** A negative exponent means take the reciprocal of the base raised to the positive exponent. $$a^{-n} = \frac{1}{a^n}$$ 7. **Fractional exponent rule:** A fractional exponent represents a root. $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$ These laws help simplify and manipulate expressions involving exponents efficiently.