1. **State the problem:** Find the product in simplest radical form for $\sqrt{88} \cdot \sqrt{24}$. 2. **Use the property of square roots:** $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. 3. **Apply the property:** $$\sqrt{88} \cdot \sqrt{24} = \sqrt{88 \times 24}$$ 4. **Calculate the product inside the radical:** $$88 \times 24 = 2112$$ 5. **Simplify $\sqrt{2112}$ by prime factorization:** $$2112 = 2^6 \times 3 \times 11$$ 6. **Rewrite the radical using factors:** $$\sqrt{2112} = \sqrt{2^6 \times 3 \times 11}$$ 7. **Extract perfect squares:** $$\sqrt{2^6} = 2^3 = 8$$ 8. **Express the simplified radical:** $$\sqrt{2112} = 8 \sqrt{3 \times 11} = 8 \sqrt{33}$$ **Final answer:** $$\boxed{8 \sqrt{33}}$$