1. The problem is to understand and apply the Law of Indices (also called the Laws of Exponents). 2. The Law of Indices states rules for simplifying expressions involving powers of the same base. Important rules include: - $a^m \times a^n = a^{m+n}$ (Product rule) - $\frac{a^m}{a^n} = a^{m-n}$ (Quotient rule) - $(a^m)^n = a^{mn}$ (Power of a power) - $a^0 = 1$ (Zero exponent rule, for $a \neq 0$) - $a^{-n} = \frac{1}{a^n}$ (Negative exponent rule) 3. For example, simplify $x^3 \times x^4$: $$x^3 \times x^4 = x^{3+4} = x^7$$ 4. Another example, simplify $\frac{y^5}{y^2}$: $$\frac{y^5}{y^2} = y^{5-2} = y^3$$ 5. Simplify $(z^2)^4$: $$ (z^2)^4 = z^{2 \times 4} = z^8$$ 6. Simplify $a^0$ (where $a \neq 0$): $$a^0 = 1$$ 7. Simplify $b^{-3}$: $$b^{-3} = \frac{1}{b^3}$$ These rules help simplify expressions with powers efficiently and correctly.