7. The expression \( \frac{5^{x+7}}{625} \) is equivalent to which of the following? \( \begin{array}{ll}\text { (1) } 5^{x+11} & \text { (3) }\left(\frac{1}{125}\right)^{x+7} \\ \text { (2) } 125^{x-4} & \text { (4) } 5^{x+3}\end{array} \)

The expression \( \frac{5^{x+7}}{625} \) can be simplified by recognizing that \( 625 \) is \( 5^4 \). So, we rewrite the expression as \( \frac{5^{x+7}}{5^4} = 5^{(x+7)-4} = 5^{x+3} \). Hence, the correct option is \( 5^{x+3} \), which is option (4)! Now, here's something interesting: exponential expressions often arise in real-life situations, like when calculating interest rates or population growth, where understanding the base can directly relate to growth factors. This connection makes exponents not just an abstract concept but a tool for understanding dynamic systems! As a quick tip, when working with expressions like this, always remember to simplify bases first. A common mistake is overlooking the simplification of numbers like \(625\) into its exponential form, which can lead to confusion. Keep your eyes peeled for these opportunities; they can save a lot of time and help you spot the correct answer quicker!