M. \( \left(\frac{1}{3}\right)^{4} \cdot\left(9^{3}\right)^{2} \)

To simplify the expression \( M = \left(\frac{1}{3}\right)^{4} \cdot\left(9^{3}\right)^{2} \), let's break it down step by step. First, evaluate \( \left(\frac{1}{3}\right)^{4} \): \[ \left(\frac{1}{3}\right)^{4} = \frac{1^{4}}{3^{4}} = \frac{1}{81} \] Next, evaluate \( \left(9^{3}\right)^{2} \): Since \( 9 = 3^{2} \), we can rewrite this as: \[ \left(9^{3}\right)^{2} = \left((3^{2})^{3}\right)^{2} = (3^{6})^{2} = 3^{12} \] Now we can combine the two results: \[ M = \frac{1}{81} \cdot 3^{12} \] Now we know \( 81 = 3^{4} \), so we can rewrite \( \frac{1}{81} \) as: \[ \frac{1}{81} = \frac{1}{3^{4}} = 3^{-4} \] Thus: \[ M = 3^{-4} \cdot 3^{12} = 3^{-4 + 12} = 3^{8} \] This gives us: \[ M = 6561 \] So, the final result is \( M = 6561 \).