1. The problem: Understand and explain the rules of exponents. 2. The basic 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: $$a^{-n} = \frac{1}{a^n}$$ - Power of a product: $$(ab)^n = a^n b^n$$ - Power of a quotient: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$ 3. Explanation: - The product rule means when multiplying powers with the same base, add the exponents. - The quotient rule means when dividing powers with the same base, subtract the exponents. - The power rule means when raising a power to another power, multiply the exponents. - Any nonzero number raised to the zero power equals 1. - A negative exponent means take the reciprocal of the base raised to the positive exponent. - When raising a product or quotient to a power, raise each factor or numerator and denominator to that power separately. 4. Example: Simplify $$2^3 \times 2^4$$ using the product rule: $$2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$$ 5. Another example: Simplify $$\frac{5^6}{5^2}$$ using the quotient rule: $$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$$ 6. Final note: These rules apply only when the bases are the same and the base is not zero when using zero or negative exponents.