An expression is shown. \[ \left(\frac{3}{4}\right)^{2} \cdot\left(\frac{3}{4}\right)^{5} \] Which of the following is equivalent to the expression shown? A \( \frac{3 \pi}{4^{7}} \) B \( \frac{3^{20}}{4^{10}} \) C \( \frac{37}{4} \) D \( \frac{3}{40} \)

When you have an expression like \(\left(\frac{3}{4}\right)^{2} \cdot\left(\frac{3}{4}\right)^{5}\), you can combine the exponents because of the properties of exponents. Specifically, \(a^m \cdot a^n = a^{m+n}\). So, here you would add the exponents: \(2 + 5 = 7\). This gives you: \[ \left(\frac{3}{4}\right)^{2} \cdot\left(\frac{3}{4}\right)^{5} = \left(\frac{3}{4}\right)^{7} = \frac{3^{7}}{4^{7}}. \] None of the answer choices directly match \(\frac{3^{7}}{4^{7}}\), but the format is crucial. It indicates that for options to be correct, they should represent this expression correctly. To explore a bit more about the expression \(\frac{3^{7}}{4^{7}}\), it's worth noting that when you see such fractions raised to the same power, they can often be broken down further in problems involving ratios or probabilities. For instance, while this isn't a direct application of fractions raised to powers, it does remind us of situations in statistics where probabilities are compounded—especially in binomial experiments! When calculating combinations or events, sometimes having a clear understanding of manipulating powers can lead to easier answers. Keep practicing those exponent rules!