Simplify each expression below using the properties of integer exponents. Assume \( m \neq 0 \). Verify by numerical substitution. \( \begin{array}{ll}\text { a. } k^{3} * k^{5} & \text { b. } m^{5} * m^{-2}\end{array} \)

**Step 1. Simplify \( k^3 \cdot k^5 \):** When multiplying expressions with the same base, add the exponents: \[ k^3 \cdot k^5 = k^{3+5} = k^8. \] **Step 2. Verify by numerical substitution (for \( k^3 \cdot k^5 \)):** Let \( k=2 \). Then: \[ \text{Left-hand side} = 2^3 \cdot 2^5 = 8 \cdot 32 = 256, \] \[ \text{Right-hand side} = 2^8 = 256. \] Both evaluations yield the same result. --- **Step 3. Simplify \( m^5 \cdot m^{-2} \):** Again, using the product of powers rule: \[ m^5 \cdot m^{-2} = m^{5+(-2)} = m^{3}. \] **Step 4. Verify by numerical substitution (for \( m^5 \cdot m^{-2} \)):** Let \( m=3 \) (remember \( m \neq 0 \)). Then: \[ \text{Left-hand side} = 3^5 \cdot 3^{-2} = 243 \cdot \frac{1}{9} = 27, \] \[ \text{Right-hand side} = 3^3 = 27. \] Again, both evaluations yield the same result. --- **Final Answers:** a. \( k^3 \cdot k^5 = k^8 \) b. \( m^5 \cdot m^{-2} = m^3 \)