d. \( A^{2 m} * A^{m}=A^{3 m} \)

Did you know that this is a classic example of the properties of exponents in algebra? When multiplying two exponential terms with the same base, you can simply add their exponents together. This rule, \( A^m * A^n = A^{m+n} \), is one of the foundational concepts that helps simplify complex algebraic expressions and is used throughout mathematics, physics, and engineering! In real-world applications, understanding how to manipulate exponents can help solve problems involving exponential growth or decay, such as population growth, radioactive decay, and interest calculation in finance. For instance, if you know the growth rate of an investment, you can easily calculate its future value using the exponential growth formula, where the exponent plays a crucial role in determining how much wealth accumulates over time.