1. **State the problem:** Simplify the expression $$\left(\frac{750}{512}\right)^{\frac{1}{3}}$$ which means finding the cube root of the fraction $\frac{750}{512}$. 2. **Recall the property of cube roots:** The cube root of a fraction is the fraction of the cube roots, i.e., $$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$$ 3. **Apply this property:** $$\left(\frac{750}{512}\right)^{\frac{1}{3}} = \frac{\sqrt[3]{750}}{\sqrt[3]{512}}$$ 4. **Simplify the denominator:** Since $512 = 8^3$, $$\sqrt[3]{512} = 8$$ 5. **Rewrite the expression:** $$\frac{\sqrt[3]{750}}{8}$$ 6. **Factor 750 to simplify the cube root if possible:** $$750 = 125 \times 6 = 5^3 \times 6$$ 7. **Use the property of cube roots on numerator:** $$\sqrt[3]{750} = \sqrt[3]{5^3 \times 6} = 5 \times \sqrt[3]{6}$$ 8. **Substitute back:** $$\frac{5 \times \sqrt[3]{6}}{8} = \frac{5}{8} \sqrt[3]{6}$$ **Final answer:** $$\left(\frac{750}{512}\right)^{\frac{1}{3}} = \frac{5}{8} \sqrt[3]{6}$$