1. The problem: Understand and apply the power laws in algebra. 2. Power laws are rules for simplifying expressions involving exponents. The main power laws are: - Product of powers: $$a^m \times a^n = a^{m+n}$$ - Quotient of powers: $$\frac{a^m}{a^n} = a^{m-n}$$ - Power of a power: $$(a^m)^n = a^{m \times n}$$ - Power of a product: $$(ab)^m = a^m b^m$$ - Power of a quotient: $$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$$ 3. Important rules: - The base $a$ must be nonzero for quotient laws. - Exponents can be any real number. - When multiplying powers with the same base, add exponents. - When dividing powers with the same base, subtract exponents. - When raising a power to another power, multiply exponents. 4. Example: Simplify $$\frac{(x^3)^2 \times x^4}{x^5}$$ Step 1: Apply power of a power: $$(x^3)^2 = x^{3 \times 2} = x^6$$ Step 2: Multiply powers with same base: $$x^6 \times x^4 = x^{6+4} = x^{10}$$ Step 3: Divide powers with same base: $$\frac{x^{10}}{x^5} = x^{10-5} = x^5$$ 5. Final answer: $$x^5$$ Power laws help simplify expressions with exponents efficiently and correctly.