1. Simplify the expression $x^3 \cdot x^{-5}$. 2. Simplify the expression $\frac{m^{-4}}{m^{-2}}$. 3. Simplify the expression $(2^{-3})^2$. 4. Simplify the expression $\frac{5^0 \cdot 5^3}{5^{-2}}$. 5. Simplify the expression $(a^{-2} b^3)(a^4 b^{-1})$. --- ### Step 1: State the problem We are asked to simplify expressions involving exponents. ### Step 2: Recall exponent rules - $a^m \cdot a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(a^m)^n = a^{m \cdot n}$ - $a^0 = 1$ for $a \neq 0$ ### Step 3: Simplify each expression 1. $x^3 \cdot x^{-5} = x^{3 + (-5)} = x^{-2} = \frac{1}{x^2}$ 2. $\frac{m^{-4}}{m^{-2}} = m^{-4 - (-2)} = m^{-4 + 2} = m^{-2} = \frac{1}{m^2}$ 3. $(2^{-3})^2 = 2^{-3 \cdot 2} = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}$ 4. $\frac{5^0 \cdot 5^3}{5^{-2}} = \frac{1 \cdot 5^3}{5^{-2}} = 5^{3 - (-2)} = 5^{3 + 2} = 5^5 = 3125$ 5. $(a^{-2} b^3)(a^4 b^{-1}) = a^{-2 + 4} b^{3 + (-1)} = a^2 b^2$ ### Final answers: 1. $\frac{1}{x^2}$ 2. $\frac{1}{m^2}$ 3. $\frac{1}{64}$ 4. $3125$ 5. $a^2 b^2$