Give your answer with positive exponent 1.1.1. \( \frac{9 x^{-1} x^{-2} y^{2}}{\left(3 x y^{-1}\right)^{-3}} \) 1.1.2. \( \frac{6\left(x^{2} y\right)\left(x^{-2} y\right)^{-1}}{3 x^{-3} y^{5}} \)

Sure, let's simplify each expression and ensure all exponents are positive. --- ### **1.1.1.** \( \frac{9 x^{-1} x^{-2} y^{2}}{\left(3 x y^{-1}\right)^{-3}} \) **Step-by-Step Simplification:** 1. **Combine the exponents in the numerator:** \[ x^{-1} \cdot x^{-2} = x^{-3} \] So, the numerator becomes: \[ 9 x^{-3} y^{2} \] 2. **Simplify the denominator:** \[ \left(3 x y^{-1}\right)^{-3} = \frac{1}{\left(3 x y^{-1}\right)^{3}} = \frac{1}{27 x^{3} y^{-3}} = \frac{y^{3}}{27 x^{3}} \] 3. **Combine the numerator and the simplified denominator:** \[ \frac{9 x^{-3} y^{2}}{\frac{y^{3}}{27 x^{3}}} = 9 x^{-3} y^{2} \times \frac{27 x^{3}}{y^{3}} = 243 x^{0} y^{-1} \] (Since \(9 \times 27 = 243\) and \(x^{-3} \times x^{3} = x^{0} = 1\)) 4. **Express with positive exponents:** \[ 243 y^{-1} = \frac{243}{y} \] **Final Answer:** \[ \frac{243}{y} \] --- ### **1.1.2.** \( \frac{6\left(x^{2} y\right)\left(x^{-2} y\right)^{-1}}{3 x^{-3} y^{5}} \) **Step-by-Step Simplification:** 1. **Simplify the expression inside the numerator:** \[ \left(x^{-2} y\right)^{-1} = x^{2} y^{-1} \] So, the numerator becomes: \[ 6 \cdot x^{2} y \cdot x^{2} y^{-1} = 6 x^{4} y^{0} = 6 x^{4} \] (Since \(y \cdot y^{-1} = y^{0} = 1\)) 2. **Simplify the entire expression:** \[ \frac{6 x^{4}}{3 x^{-3} y^{5}} = \frac{6}{3} \cdot x^{4 - (-3)} \cdot y^{-5} = 2 x^{7} y^{-5} \] (Since \(6 \div 3 = 2\), \(x^{4} \times x^{3} = x^{7}\)) 3. **Express with positive exponents:** \[ 2 x^{7} y^{-5} = \frac{2 x^{7}}{y^{5}} \] **Final Answer:** \[ \frac{2 x^{7}}{y^{5}} \] ---