1. **State the problem:** Simplify the expression $\sqrt{27} \times \sqrt{80}$.\n\n2. **Recall the property of square roots:** $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$. This means we can combine the roots under one radical.\n\n3. **Apply the property:** $$\sqrt{27} \times \sqrt{80} = \sqrt{27 \times 80}$$\n\n4. **Multiply inside the square root:** $$27 \times 80 = 2160$$ So, $$\sqrt{27} \times \sqrt{80} = \sqrt{2160}$$\n\n5. **Simplify $\sqrt{2160}$ by prime factorization:** $$2160 = 2^4 \times 3^3 \times 5$$ \n6. **Rewrite the square root using factors:** $$\sqrt{2160} = \sqrt{2^4 \times 3^3 \times 5} = \sqrt{(2^4)} \times \sqrt{(3^3)} \times \sqrt{5}$$\n\n7. **Simplify each square root:** $$\sqrt{2^4} = 2^{4/2} = 2^2 = 4$$ $$\sqrt{3^3} = \sqrt{3^2 \times 3} = 3 \times \sqrt{3}$$ $$\sqrt{5} = \sqrt{5}$$ \n8. **Combine the simplified parts:** $$4 \times 3 \times \sqrt{3} \times \sqrt{5} = 12 \times \sqrt{15}$$ \n9. **Final answer:** $$\sqrt{27} \times \sqrt{80} = 12 \sqrt{15}$$