1. **State the problem:** Simplify the cube root of the fraction $\frac{27}{8}$, written as $\sqrt[3]{\frac{27}{8}}$. 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 the property:** $$\sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}}$$ 4. **Evaluate the cube roots:** - $\sqrt[3]{27} = 3$ because $3^3 = 27$ - $\sqrt[3]{8} = 2$ because $2^3 = 8$ 5. **Write the simplified fraction:** $$\frac{3}{2}$$ 6. **Check the answer choices:** The simplified form $\frac{3}{2}$ matches the third option. **Final answer:** $\frac{3}{2}$