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

To simplify the expressions, we can use the properties of exponents. For part (a), \( k^{3} * k^{5} \) can be simplified using the rule that states when you multiply like bases, you add the exponents: \[ k^{3} * k^{5} = k^{3 + 5} = k^{8}. \] For part (b), \( m^{5} * m^{-2} \) also follows this rule, but here we need to remember that a negative exponent represents a reciprocal: \[ m^{5} * m^{-2} = m^{5 + (-2)} = m^{3}. \] Now, let's verify both parts with numerical substitution. If we let \( k = 2 \) and \( m = 4 \): For (a): \[ k^{3} * k^{5} = 2^{3} * 2^{5} = 8 * 32 = 256, \] and \[ k^{8} = 2^{8} = 256. \] For (b): \[ m^{5} * m^{-2} = 4^{5} * 4^{-2} = 1024 * \frac{1}{16} = 64, \] and \[ m^{3} = 4^{3} = 64. \] Both simplifications match their respective numerical values!