Which is an equivalent expression for \( \left(3^{2} \cdot 5^{4}\right)^{3} ? \) \( 3^{2} \cdot 5^{12} \) \( 3^{5} \cdot 5^{7} \) \( 3^{6} \cdot 5^{12} \) \( 15^{18} \)

To find an equivalent expression for \( \left(3^{2} \cdot 5^{4}\right)^{3} \), we use the power of a product rule. This means we can distribute the exponent across both terms within the parentheses. So, we have: \[ \left(3^{2}\right)^{3} \cdot \left(5^{4}\right)^{3} = 3^{6} \cdot 5^{12}. \] Therefore, the correct choice from the given options is \( 3^{6} \cdot 5^{12} \). Now, here’s some extra fun for you! Did you know that exponents are all around us? They can represent exponential growth, for instance, like populations or investments growing over time! If you've ever wondered how quickly money can multiply in a compound interest account, it’s all about those exponent rules. Another cool tidbit: while working with exponents, a common mistake is forgetting to distribute the exponent correctly. Always remember, \( (a \cdot b)^{c} \) is not the same as \( a^{c} + b^{c} \); it’s multiplied! So keep that in mind next time you're crunching numbers!