\( \begin{array}{ll}\text { (b) }\left(-2 x^{3} y^{2}\right)^{3} & \text { (c) }\left(3^{3}\right)^{2} \cdot 3^{2} \\ \text { (e) }\left(\frac{-12 x^{3}}{15 y^{-2}}\right)^{2} & \text { (f) }\left(\frac{10 y^{4}}{4 y^{6}}\right)^{-2}\end{array} \)

Let's simplify each of the given expressions step by step. --- ### **(b) \(\left(-2 x^{3} y^{2}\right)^{3}\)** **Step-by-Step Simplification:** 1. **Apply the exponent to each factor inside the parentheses:** \[ \left(-2\right)^3 \cdot \left(x^{3}\right)^3 \cdot \left(y^{2}\right)^3 \] 2. **Calculate each part:** - \(\left(-2\right)^3 = -8\) - \(\left(x^{3}\right)^3 = x^{9}\) (since \(3 \times 3 = 9\)) - \(\left(y^{2}\right)^3 = y^{6}\) (since \(2 \times 3 = 6\)) 3. **Combine the results:** \[ -8 x^{9} y^{6} \] **Final Answer:** \[ -8 x^{9} y^{6} \] --- ### **(c) \(\left(3^{3}\right)^{2} \cdot 3^{2}\)** **Step-by-Step Simplification:** 1. **Simplify \(\left(3^{3}\right)^{2}\):** \[ 3^{3 \times 2} = 3^{6} \] 2. **Multiply by \(3^{2}\):** \[ 3^{6} \cdot 3^{2} = 3^{6+2} = 3^{8} \] 3. **Calculate \(3^{8}\) if desired:** \[ 3^{8} = 6561 \] **Final Answer:** \[ 3^{8} \quad \text{or} \quad 6561 \] --- ### **(e) \(\left(\frac{-12 x^{3}}{15 y^{-2}}\right)^{2}\)** **Step-by-Step Simplification:** 1. **Simplify the expression inside the parentheses:** \[ \frac{-12 x^{3}}{15 y^{-2}} = \frac{-12 x^{3} \cdot y^{2}}{15} = \frac{-4 x^{3} y^{2}}{5} \quad (\text{simplifying } -12/15 = -4/5) \] 2. **Apply the exponent to each factor:** \[ \left(\frac{-4 x^{3} y^{2}}{5}\right)^{2} = \frac{(-4)^2 \cdot (x^{3})^2 \cdot (y^{2})^2}{5^2} \] 3. **Calculate each part:** - \((-4)^2 = 16\) - \((x^{3})^2 = x^{6}\) - \((y^{2})^2 = y^{4}\) - \(5^2 = 25\) 4. **Combine the results:** \[ \frac{16 x^{6} y^{4}}{25} \] **Final Answer:** \[ \frac{16 x^{6} y^{4}}{25} \] --- ### **(f) \(\left(\frac{10 y^{4}}{4 y^{6}}\right)^{-2}\)** **Step-by-Step Simplification:** 1. **Simplify the expression inside the parentheses:** \[ \frac{10 y^{4}}{4 y^{6}} = \frac{10}{4} \cdot y^{4-6} = \frac{5}{2} \cdot y^{-2} \] 2. **Apply the exponent \(-2\) to the entire expression:** \[ \left(\frac{5}{2} \cdot y^{-2}\right)^{-2} = \left(\frac{5}{2}\right)^{-2} \cdot \left(y^{-2}\right)^{-2} \] 3. **Calculate each part:** - \(\left(\frac{5}{2}\right)^{-2} = \left(\frac{2}{5}\right)^{2} = \frac{4}{25}\) - \(\left(y^{-2}\right)^{-2} = y^{4}\) (since \(-2 \times -2 = 4\)) 4. **Combine the results:** \[ \frac{4 y^{4}}{25} \] **Final Answer:** \[ \frac{4 y^{4}}{25} \] --- **Summary of Simplified Expressions:** - **(b)** \(-8 x^{9} y^{6}\) - **(c)** \(3^{8}\) or \(6561\) - **(e)** \(\dfrac{16 x^{6} y^{4}}{25}\) - **(f)** \(\dfrac{4 y^{4}}{25}\)