1. **Problem:** Simplify the expression $4\sqrt{3} \cdot 2\sqrt{3} - 5\sqrt{27}$. 2. **Recall the rules:** - Multiplying square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. - Simplify square roots by factoring out perfect squares. - Multiply coefficients separately from radicals. 3. **Step-by-step solution:** - Multiply $4\sqrt{3}$ and $2\sqrt{3}$: $$4\sqrt{3} \cdot 2\sqrt{3} = (4 \cdot 2)(\sqrt{3} \cdot \sqrt{3}) = 8 \cdot \sqrt{9} = 8 \cdot 3 = 24$$ - Simplify $5\sqrt{27}$: $$\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}$$ So, $$5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3}$$ - Combine the terms: $$24 - 15\sqrt{3}$$ 4. **Final answer:** $$24 - 15\sqrt{3}$$