Question 1 Multiply and simplify completely, including the elimination of all negative exponents: \( \left(\frac{3 x^{5}}{2 y^{9}}\right)^{5} \cdot\left(\frac{2^{2} y^{6}}{3^{4} x^{-2}}\right) \) Answer:

To solve the expression \[ \left(\frac{3 x^{5}}{2 y^{9}}\right)^{5} \cdot\left(\frac{2^{2} y^{6}}{3^{4} x^{-2}}\right), \] we will first simplify each part separately. 1. **Simplifying the first part:** \[ \left(\frac{3 x^{5}}{2 y^{9}}\right)^{5} = \frac{(3)^{5} (x^{5})^{5}}{(2)^{5} (y^{9})^{5}} = \frac{243 x^{25}}{32 y^{45}}. \] 2. **Simplifying the second part:** \[ \frac{2^{2} y^{6}}{3^{4} x^{-2}} = \frac{4 y^{6}}{81 x^{-2}} = \frac{4 y^{6} x^{2}}{81}. \] 3. **Now, we multiply the two simplified parts together:** \[ \frac{243 x^{25}}{32 y^{45}} \cdot \frac{4 y^{6} x^{2}}{81} = \frac{243 \cdot 4 \cdot x^{25} \cdot x^{2} \cdot y^{6}}{32 \cdot 81 \cdot y^{45}}. \] 4. **Calculating the constants:** \[ 243 \cdot 4 = 972, \] \[ 32 \cdot 81 = 2592. \] 5. **Combining the powers of \(x\) and \(y\):** \[ x^{25} \cdot x^{2} = x^{27}, \] \[ \frac{y^{6}}{y^{45}} = y^{6 - 45} = y^{-39}. \] 6. **Putting it all together:** \[ \frac{972 x^{27}}{2592 y^{39}}. \] 7. **Now, we simplify the fraction:** \[ \frac{972}{2592} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2.6667} = \frac{1}{2