1. **State the problem:** Simplify the expression $$\sqrt{x} \cdot \sqrt{4x} + \sqrt{100x^2} - \sqrt{9} \sqrt{x}$$. 2. **Recall the properties of square roots:** - $$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$ - $$\sqrt{a^2} = |a|$$ (absolute value of a) 3. **Simplify each term:** - $$\sqrt{x} \cdot \sqrt{4x} = \sqrt{4x^2} = 2|x|$$ - $$\sqrt{100x^2} = 10|x|$$ - $$\sqrt{9} \sqrt{x} = 3\sqrt{x}$$ 4. **Rewrite the expression:** $$2|x| + 10|x| - 3\sqrt{x}$$ 5. **Combine like terms:** $$12|x| - 3\sqrt{x}$$ 6. **Final answer:** $$12|x| - 3\sqrt{x}$$ This is the simplified form of the original expression, assuming $$x \geq 0$$ so that $$\sqrt{x}$$ is defined and $$|x| = x$$.