2.2. Simplify the following: \( 2.2 .1 \quad 4 x^{2} y^{-4} \times\left(\frac{2 y^{3}}{x^{3}}\right)^{3} \)

We begin with the expression: \[ 4x^2 y^{-4} \times \left(\frac{2y^3}{x^3}\right)^3. \] **Step 1. Expand the exponent.** Apply the exponent \(3\) to both numerator and denominator: \[ \left(\frac{2y^3}{x^3}\right)^3 = \frac{(2y^3)^3}{(x^3)^3}. \] This becomes: \[ \frac{2^3 \, (y^3)^3}{x^{9}} = \frac{8 \, y^9}{x^9}. \] **Step 2. Multiply the expressions.** The original expression now is: \[ 4x^2 y^{-4} \times \frac{8y^9}{x^9}. \] Multiply the coefficients, the \(x\)-terms, and the \(y\)-terms separately: \[ \text{Coefficient: } 4 \times 8 = 32. \] \[ \text{\(x\)-terms: } x^2 \times \frac{1}{x^9} = x^{2-9} = x^{-7}. \] \[ \text{\(y\)-terms: } y^{-4} \times y^9 = y^{-4+9} = y^5. \] Thus, the expression simplifies to: \[ 32\, x^{-7}\, y^5. \] **Step 3. Write the final answer using positive exponents.** Since \(x^{-7} = \frac{1}{x^7}\), the final simplified form is: \[ \frac{32\, y^5}{x^7}. \]