1. The problem is to understand the law of indices, which are rules for simplifying expressions involving powers or exponents. 2. The main laws of indices are: - Product rule: $$a^m \times a^n = a^{m+n}$$ - Quotient rule: $$\frac{a^m}{a^n} = a^{m-n}$$ - Power of a power: $$(a^m)^n = a^{mn}$$ - 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: $$a^0 = 1$$ (where $a \neq 0$) - Negative exponent: $$a^{-m} = \frac{1}{a^m}$$ 3. These rules help simplify expressions with exponents by combining or breaking down powers. 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 of a power rule: $$3^{2 \times 4} = 3^8 = 6561$$ 7. Remember these laws only apply when the base is the same and the base is not zero when using zero or negative exponents. Understanding these laws will help you work with powers efficiently.