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

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UpStudy AI · Accepted Answer

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 $$ \dfrac{q^{21}}{250} $$.
The simplified expression is $$ \frac{q^{21}}{250} $$.

b)

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