1. **State the problem:** Simplify the expression $$\sqrt[3]{64 x^{28} y^{17} z^{4}}$$. 2. **Recall the cube root property:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{a b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$. 3. **Apply the cube root to each factor:** $$\sqrt[3]{64} \cdot \sqrt[3]{x^{28}} \cdot \sqrt[3]{y^{17}} \cdot \sqrt[3]{z^{4}}$$ 4. **Simplify each cube root:** - Since $$64 = 4^3$$, $$\sqrt[3]{64} = 4$$. - For $$x^{28}$$, use $$\sqrt[3]{x^{28}} = x^{\frac{28}{3}} = x^{9 + \frac{1}{3}} = x^{9} x^{\frac{1}{3}}$$. - For $$y^{17}$$, $$\sqrt[3]{y^{17}} = y^{\frac{17}{3}} = y^{5 + \frac{2}{3}} = y^{5} y^{\frac{2}{3}}$$. - For $$z^{4}$$, $$\sqrt[3]{z^{4}} = z^{\frac{4}{3}} = z^{1 + \frac{1}{3}} = z^{1} z^{\frac{1}{3}}$$. 5. **Rewrite the expression:** $$4 \cdot x^{9} x^{\frac{1}{3}} \cdot y^{5} y^{\frac{2}{3}} \cdot z^{1} z^{\frac{1}{3}}$$ 6. **Group integer powers and fractional powers:** $$4 x^{9} y^{5} z^{1} \cdot x^{\frac{1}{3}} y^{\frac{2}{3}} z^{\frac{1}{3}}$$ 7. **Express the fractional powers back under a cube root:** $$4 x^{9} y^{5} z \cdot \sqrt[3]{x y^{2} z}$$ **Final answer:** $$4 x^{9} y^{5} z \sqrt[3]{x y^{2} z}$$