1. **State the problem:** Simplify the expression $$(4 \times \sqrt{18x}) (6 \sqrt{6x})$$. 2. **Recall the properties of square roots and multiplication:** - $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ - Multiplication is associative and commutative, so we can rearrange terms. 3. **Rewrite the expression:** $$4 \times 6 \times \sqrt{18x} \times \sqrt{6x}$$ 4. **Multiply the constants:** $$4 \times 6 = 24$$ 5. **Multiply the square roots:** $$\sqrt{18x} \times \sqrt{6x} = \sqrt{(18x)(6x)} = \sqrt{108x^2}$$ 6. **Simplify inside the square root:** $$108x^2 = 36 \times 3 \times x^2$$ 7. **Use the property $\sqrt{a b} = \sqrt{a} \times \sqrt{b}$:** $$\sqrt{108x^2} = \sqrt{36} \times \sqrt{3} \times \sqrt{x^2}$$ 8. **Simplify each square root:** $$\sqrt{36} = 6, \quad \sqrt{3} \text{ stays as is}, \quad \sqrt{x^2} = x$$ 9. **Combine the simplified parts:** $$6 \times \sqrt{3} \times x = 6x \sqrt{3}$$ 10. **Multiply by the constant 24:** $$24 \times 6x \sqrt{3} = 144x \sqrt{3}$$ **Final answer:** $$\boxed{144x \sqrt{3}}$$