(b) Simplify each of the following expressions: (i) \( \left(\frac{x}{y}\right)^{4}\left(y^{2} x\right)^{3} \) \( \square \)

To simplify the expression \( \left(\frac{x}{y}\right)^{4}\left(y^{2} x\right)^{3} \), start by applying the power of a quotient and product of powers. First, break it down: \[ \left(\frac{x}{y}\right)^{4} = \frac{x^{4}}{y^{4}} \] and \[ \left(y^{2} x\right)^{3} = y^{2 \cdot 3} x^{3} = y^{6} x^{3}. \] Now, combine these results: \[ \frac{x^{4}}{y^{4}} \cdot (y^{6} x^{3}) = \frac{x^{4} x^{3} y^{6}}{y^{4}} = \frac{x^{7} y^{6}}{y^{4}}. \] Finally, simplify: \[ \frac{x^{7} y^{6}}{y^{4}} = x^{7} y^{6-4} = x^{7} y^{2}. \] So the simplified expression is \( x^{7} y^{2} \).