Simplify express your answers with positive exponenets. \( \left(\frac{2 c^{5} b^{2}}{12 c^{-2} b}\right)^{3} \)

To simplify \( \left(\frac{2 c^{5} b^{2}}{12 c^{-2} b}\right)^{3} \), we first simplify the expression inside the parentheses: 1. Simplify the fraction: \[ \frac{2}{12} = \frac{1}{6}, \quad \text{thus } \frac{2 c^{5} b^{2}}{12 c^{-2} b} = \frac{1}{6} \cdot \frac{c^{5} b^{2}}{c^{-2} b} \] \[ = \frac{1}{6} \cdot c^{5 - (-2)} \cdot b^{2 - 1} = \frac{1}{6} \cdot c^{5 + 2} \cdot b^{2 - 1} = \frac{1}{6} \cdot c^{7} \cdot b^{1} \] 2. Now our expression is: \[ \left( \frac{1}{6} c^{7} b \right)^{3} \] 3. Applying the exponent: \[ = \left(\frac{1}{6}\right)^{3} \cdot c^{7 \cdot 3} \cdot b^{3} \] \[ = \frac{1}{216} \cdot c^{21} \cdot b^{3} \] Lastly, our simplified expression is: \[ \frac{c^{21} b^{3}}{216} \]