1. **State the problem:** Simplify the expression $$\sqrt[3]{108c^{17}}$$. 2. **Recall the formula and rules:** - The cube root of a product is the product of the cube roots: $$\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}$$. - For powers, $$\sqrt[3]{c^n} = c^{\frac{n}{3}}$$. - We can separate the exponent into an integer part and a fractional part: $$c^{\frac{n}{3}} = c^{k + r} = c^k \times c^r$$ where $$k = \lfloor \frac{n}{3} \rfloor$$ and $$r = \frac{n}{3} - k$$. 3. **Simplify the numerical part:** - Factor 108: $$108 = 27 \times 4$$. - Since $$\sqrt[3]{27} = 3$$, we have $$\sqrt[3]{108} = \sqrt[3]{27 \times 4} = 3 \sqrt[3]{4}$$. 4. **Simplify the variable part:** - $$\sqrt[3]{c^{17}} = c^{\frac{17}{3}} = c^{5 + \frac{2}{3}} = c^5 \times c^{\frac{2}{3}} = c^5 \sqrt[3]{c^2}$$. 5. **Combine the simplified parts:** $$\sqrt[3]{108c^{17}} = 3 \sqrt[3]{4} \times c^5 \sqrt[3]{c^2} = 3 c^5 \sqrt[3]{4 c^2}$$. 6. **Final answer:** $$3 c^5 \sqrt[3]{4 c^2}$$. This matches option 4 in the multiple-choice list.