1. The problem: Understand and explain the rules of exponents. 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: $$a^{-n} = \frac{1}{a^n}$$ - 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: - 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. - Negative exponents mean take the reciprocal of the base raised to the positive exponent. - When raising a product or quotient to a power, raise each factor to that power. 4. Examples: - $$2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$$ - $$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$$ - $$(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$$ - $$7^0 = 1$$ - $$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$$ - $$(2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36$$ - $$\left(\frac{10}{2}\right)^3 = \frac{10^3}{2^3} = \frac{1000}{8} = 125$$ These rules help simplify expressions involving exponents efficiently and correctly.