1. **State the problem:** Simplify the expression $$\sqrt{8x^7y^8}$$ assuming $$x \geq 0$$. 2. **Recall the rule:** $$\sqrt{a b} = \sqrt{a} \times \sqrt{b}$$ and $$\sqrt{x^{2n}} = x^n$$ for $$x \geq 0$$. 3. **Rewrite the radicand:** $$8x^7y^8 = 8 \times x^7 \times y^8$$. 4. **Express 8 as $$4 \times 2$$:** $$\sqrt{8x^7y^8} = \sqrt{4 \times 2 \times x^7 \times y^8}$$. 5. **Separate perfect squares:** $$\sqrt{4} = 2$$, $$\sqrt{y^8} = y^4$$, and for $$x^7$$, write as $$x^6 \times x$$. 6. **Apply the square root:** $$\sqrt{4 \times x^6 \times y^8 \times 2 \times x} = \sqrt{4} \times \sqrt{x^6} \times \sqrt{y^8} \times \sqrt{2x}$$. 7. **Simplify each:** $$\sqrt{4} = 2$$, $$\sqrt{x^6} = x^3$$, $$\sqrt{y^8} = y^4$$. 8. **Combine:** $$2 \times x^3 \times y^4 \times \sqrt{2x} = 2x^3y^4\sqrt{2x}$$. **Final answer:** $$2x^3y^4\sqrt{2x}$$. This matches the third option.