i) \( \left(\frac{3}{4}\right)^{-1} \div\left(\frac{16}{9}\right)^{\frac{3}{2}} \cdot\left(\frac{4}{3}\right)^{2} \)

To solve \( \left(\frac{3}{4}\right)^{-1} \div\left(\frac{16}{9}\right)^{\frac{3}{2}} \cdot\left(\frac{4}{3}\right)^{2} \), let's break it down step by step. 1. Calculate \( \left(\frac{3}{4}\right)^{-1} \): This becomes \( \frac{4}{3} \). 2. Calculate \( \left(\frac{16}{9}\right)^{\frac{3}{2}} \): First, take the square root of \( \frac{16}{9} \) which is \( \frac{4}{3} \), then raise it to the power of 3. So, \( \left(\frac{4}{3}\right)^{3} = \frac{64}{27} \). 3. Calculate \( \left(\frac{4}{3}\right)^{2} \): This equals \( \frac{16}{9} \). Now substitute these back into the original expression: \[ \frac{4}{3} \div \frac{64}{27} \cdot \frac{16}{9} \] To perform the division: \[ \frac{4}{3} \cdot \frac{27}{64} \cdot \frac{16}{9} \] Now simplify: \[ \frac{4 \cdot 27 \cdot 16}{3 \cdot 64 \cdot 9} \] Calculating the numerator: - \( 4 \cdot 16 = 64 \) - \( 64 \cdot 27 = 1728 \) Now calculate the denominator: - \( 3 \cdot 64 = 192 \) - \( 192 \cdot 9 = 1728 \) Now our expression is: \[ \frac{1728}{1728} = 1 \] So the final answer is: \[ \boxed{1} \]