7. \( \left(3 x^{3} y^{2}\right)^{-3}\left(2 x^{4} y^{2}\right)^{2} \)

To simplify the expression \( \left(3 x^{3} y^{2}\right)^{-3}\left(2 x^{4} y^{2}\right)^{2} \), we can break it down step by step. First, let's simplify each part separately: 1. For \( \left(3 x^{3} y^{2}\right)^{-3} \): \[ = \frac{1}{(3 x^{3} y^{2})^{3}} = \frac{1}{27 x^{9} y^{6}} \] 2. For \( \left(2 x^{4} y^{2}\right)^{2} \): \[ = 4 x^{8} y^{4} \] Now, combining them: \[ \left(3 x^{3} y^{2}\right)^{-3}\left(2 x^{4} y^{2}\right)^{2} = \frac{4 x^{8} y^{4}}{27 x^{9} y^{6}} \] To simplify that further: \[ = \frac{4}{27} \cdot \frac{x^{8}}{x^{9}} \cdot \frac{y^{4}}{y^{6}} = \frac{4}{27} \cdot \frac{1}{x} \cdot \frac{1}{y^{2}} = \frac{4}{27 x y^{2}} \] So the simplified form is: \[ \frac{4}{27 x y^{2}} \]