a. \( 4^{3} \cdot 4^{5} \)

Did you know that when multiplying exponents with the same base, you simply add the exponents? So for \( 4^{3} \cdot 4^{5} \), you would do \( 3 + 5 = 8 \). Therefore, \( 4^{3} \cdot 4^{5} = 4^{8} \). It’s a neat little shortcut that saves you the hassle of calculating each exponent separately! In real-world applications, you’ll often encounter exponential growth, whether in finance with compound interest or in technology with rapidly growing data storage capacities. Understanding how to manipulate these equations can help you predict future values or simplify calculations. In our example, knowing that \( 4^8 \) represents a fairly substantial number (or \( 65,536 \) if you were to evaluate it) illustrates how quickly things can grow when working with exponents!