1. **State the problem:** We need to find the area of a rectangle with length $2\sqrt{27}$ cm and width $4\sqrt{3}$ cm. 2. **Formula for area of a rectangle:** $$\text{Area} = \text{length} \times \text{width}$$ 3. **Substitute the given values:** $$\text{Area} = 2\sqrt{27} \times 4\sqrt{3}$$ 4. **Multiply the constants and the radicals separately:** $$= (2 \times 4) \times (\sqrt{27} \times \sqrt{3}) = 8 \times \sqrt{27 \times 3}$$ 5. **Simplify inside the square root:** $$\sqrt{27 \times 3} = \sqrt{81}$$ 6. **Calculate the square root:** $$\sqrt{81} = 9$$ 7. **Multiply to find the area:** $$8 \times 9 = 72$$ **Final answer:** The area of the rectangle is $72$ cm$^2$.