1. **State the problem:** Simplify the expression $$7\sqrt{18x^2} + \sqrt{48x^4} + \sqrt{x^8}$$. 2. **Recall the square root properties:** - $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ - $$\sqrt{x^n} = x^{\frac{n}{2}}$$ for $$x \geq 0$$. 3. **Simplify each term:** - $$7\sqrt{18x^2} = 7 \cdot \sqrt{18} \cdot \sqrt{x^2} = 7 \cdot \sqrt{9 \cdot 2} \cdot x = 7 \cdot 3 \cdot \sqrt{2} \cdot x = 21x\sqrt{2}$$ - $$\sqrt{48x^4} = \sqrt{48} \cdot \sqrt{x^4} = \sqrt{16 \cdot 3} \cdot x^2 = 4x^2\sqrt{3}$$ - $$\sqrt{x^8} = x^{\frac{8}{2}} = x^4$$ 4. **Rewrite the expression:** $$21x\sqrt{2} + 4x^2\sqrt{3} + x^4$$ 5. **Final answer:** The simplified form is $$21x\sqrt{2} + 4x^2\sqrt{3} + x^4$$. This expression cannot be simplified further because the terms contain different radicals and powers of $$x$$.