1. Let's start by stating the problem: We want to understand the basic rules of exponents, which help us simplify expressions involving powers. 2. The main exponent rules are: - Product rule: $a^m \times a^n = a^{m+n}$ - Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$ - Power rule: $(a^m)^n = a^{m \times n}$ - Zero exponent rule: $a^0 = 1$ (for $a \neq 0$) - Negative exponent rule: $a^{-n} = \frac{1}{a^n}$ 3. Explanation: - When multiplying powers with the same base, add the exponents. - When dividing powers with the same base, subtract the exponents. - When raising a power to another power, multiply the exponents. - Any nonzero number raised to the zero power is 1. - A negative exponent means take the reciprocal of the base raised to the positive exponent. 4. Example: Simplify $2^3 \times 2^4$ Using the product rule: $2^{3+4} = 2^7 = 128$ 5. Example: Simplify $\frac{5^6}{5^2}$ Using the quotient rule: $5^{6-2} = 5^4 = 625$ 6. Example: Simplify $(3^2)^4$ Using the power rule: $3^{2 \times 4} = 3^8 = 6561$ These rules make working with exponents easier and help simplify expressions efficiently.