1. **State the problem:** Simplify the expression $$\sqrt{12x} - \sqrt{75x} + \sqrt{147x}$$. 2. **Recall the rule:** The square root of a product can be written as the product of square roots: $$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$$. 3. **Simplify each term by factoring out perfect squares:** - $$\sqrt{12x} = \sqrt{4 \times 3 \times x} = \sqrt{4} \times \sqrt{3x} = 2\sqrt{3x}$$ - $$\sqrt{75x} = \sqrt{25 \times 3 \times x} = \sqrt{25} \times \sqrt{3x} = 5\sqrt{3x}$$ - $$\sqrt{147x} = \sqrt{49 \times 3 \times x} = \sqrt{49} \times \sqrt{3x} = 7\sqrt{3x}$$ 4. **Rewrite the expression with simplified terms:** $$2\sqrt{3x} - 5\sqrt{3x} + 7\sqrt{3x}$$ 5. **Combine like terms:** $$ (2 - 5 + 7) \sqrt{3x} = 4\sqrt{3x}$$ **Final answer:** $$4\sqrt{3x}$$