1. The problem is to simplify expressions using index laws. 2. Index laws (or laws of exponents) include: - $a^m \times a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(a^m)^n = a^{mn}$ - $a^0 = 1$ (for $a \neq 0$) - $a^{-n} = \frac{1}{a^n}$ 3. Let's apply these laws step-by-step to simplify an example expression: $\frac{x^5 \times x^3}{x^4}$. 4. Using the multiplication law for the numerator: $x^5 \times x^3 = x^{5+3} = x^8$. 5. Now the expression is $\frac{x^8}{x^4}$. 6. Using the division law: $\frac{x^8}{x^4} = x^{8-4} = x^4$. 7. The simplified expression is $x^4$. 8. This shows how index laws help simplify expressions by adding or subtracting exponents when multiplying or dividing like bases.