1. **State the problem:** Simplify the expression $$\left(\frac{cd^8}{2c^{\frac{5}{6}} d^5}\right)^6$$ assuming all variables are positive. 2. **Recall the rules:** - When dividing like bases, subtract exponents: $$a^m / a^n = a^{m-n}$$. - When raising a power to another power, multiply exponents: $$(a^m)^n = a^{mn}$$. - Keep all exponents positive. 3. **Simplify inside the parentheses first:** $$\frac{cd^8}{2c^{\frac{5}{6}} d^5} = \frac{c^{1} d^{8}}{2 c^{\frac{5}{6}} d^{5}} = \frac{1}{2} c^{1 - \frac{5}{6}} d^{8 - 5} = \frac{1}{2} c^{\frac{1}{6}} d^{3}$$ 4. **Now raise the entire expression to the 6th power:** $$\left(\frac{1}{2} c^{\frac{1}{6}} d^{3}\right)^6 = \left(\frac{1}{2}\right)^6 \left(c^{\frac{1}{6}}\right)^6 \left(d^{3}\right)^6$$ 5. **Calculate each part:** - $$\left(\frac{1}{2}\right)^6 = \frac{1}{2^6} = \frac{1}{64}$$ - $$\left(c^{\frac{1}{6}}\right)^6 = c^{\frac{1}{6} \times 6} = c^{1} = c$$ - $$\left(d^{3}\right)^6 = d^{3 \times 6} = d^{18}$$ 6. **Combine all parts:** $$\frac{1}{64} \times c \times d^{18} = \frac{c d^{18}}{64}$$ **Final answer:** $$\boxed{\frac{c d^{18}}{64}}$$