1. **Stating the problem:** We want to understand the power laws and laws of exponents, which are rules that help us simplify expressions involving powers. 2. **Key exponent laws:** - 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: $$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$$ - Zero exponent: $$a^0 = 1$$ (for $a \neq 0$) - Negative exponent: $$a^{-m} = \frac{1}{a^m}$$ 3. **Example:** Simplify $$\frac{(x^3)^2 \times x^{-4}}{x^2}$$ 4. **Step-by-step simplification:** - Apply power of a power: $$(x^3)^2 = x^{3 \times 2} = x^6$$ - Substitute back: $$\frac{x^6 \times x^{-4}}{x^2}$$ - Use product of powers in numerator: $$x^{6 + (-4)} = x^2$$ - Now expression is $$\frac{x^2}{x^2}$$ - Use quotient of powers: $$x^{2 - 2} = x^0$$ - Apply zero exponent rule: $$x^0 = 1$$ 5. **Final answer:** The expression simplifies to 1. These laws allow us to manipulate and simplify expressions with exponents easily.