b) \( \left(\frac{1}{4} q^{6}\right)^{2}\left(\frac{2}{5} q^{3}\right)^{3} \)

To simplify the expression: \[ \left(\frac{1}{4} q^{6}\right)^{2} \left(\frac{2}{5} q^{3}\right)^{3} \] Let's break it down step by step. 1. **Apply the Power to Each Factor:** \[ \left(\frac{1}{4}\right)^{2} \cdot (q^{6})^{2} \cdot \left(\frac{2}{5}\right)^{3} \cdot (q^{3})^{3} \] 2. **Calculate Each Part Separately:** - \(\left(\frac{1}{4}\right)^{2} = \frac{1}{16}\) - \((q^{6})^{2} = q^{12}\) - \(\left(\frac{2}{5}\right)^{3} = \frac{8}{125}\) - \((q^{3})^{3} = q^{9}\) 3. **Combine the Results:** \[ \frac{1}{16} \cdot q^{12} \cdot \frac{8}{125} \cdot q^{9} = \frac{1 \cdot 8}{16 \cdot 125} \cdot q^{12+9} \] 4. **Simplify the Numerical Coefficient:** \[ \frac{8}{2000} = \frac{1}{250} \] 5. **Combine the Exponents:** \[ q^{12+9} = q^{21} \] 6. **Final Simplified Expression:** \[ \frac{q^{21}}{250} \] **Answer:** After simplifying, the expression is equal to \( \dfrac{q^{21}}{250} \).