1. **State the problem:** Simplify the expression $$K = \left(\sqrt{\sqrt{2} \sqrt{\sqrt{2}} \sqrt{\sqrt{\sqrt{2}}} \sqrt{2 \sqrt{2} \sqrt{2} \sqrt{2}}}\right)^8$$. 2. **Rewrite each nested radical in terms of powers of 2:** - $$\sqrt{2} = 2^{\frac{1}{2}}$$ - $$\sqrt{\sqrt{2}} = \sqrt{2^{\frac{1}{2}}} = 2^{\frac{1}{4}}$$ - $$\sqrt{\sqrt{\sqrt{2}}} = \sqrt{2^{\frac{1}{4}}} = 2^{\frac{1}{8}}$$ - $$\sqrt{2 \sqrt{2} \sqrt{2} \sqrt{2}} = \sqrt{2 \times 2^{\frac{1}{2}} \times 2^{\frac{1}{2}} \times 2^{\frac{1}{2}}}$$ 3. **Simplify the product inside the last radical:** - Multiply the powers of 2: $$2^{1} \times 2^{\frac{1}{2}} \times 2^{\frac{1}{2}} \times 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}} = 2^{2.5} = 2^{\frac{5}{2}}$$ - So, $$\sqrt{2 \sqrt{2} \sqrt{2} \sqrt{2}} = \sqrt{2^{\frac{5}{2}}} = 2^{\frac{5}{4}}$$ 4. **Combine all terms inside the big square root:** $$\sqrt{2^{\frac{1}{2}} \times 2^{\frac{1}{4}} \times 2^{\frac{1}{8}} \times 2^{\frac{5}{4}}} = \sqrt{2^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{5}{4}}}$$ 5. **Sum the exponents inside the square root:** $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{5}{4} = \frac{4}{8} + \frac{2}{8} + \frac{1}{8} + \frac{10}{8} = \frac{17}{8}$$ 6. **So the expression inside the big square root is:** $$\sqrt{2^{\frac{17}{8}}} = 2^{\frac{17}{16}}$$ 7. **Now the entire expression inside parentheses is:** $$2^{\frac{17}{16}}$$ 8. **Raise this to the 8th power:** $$\left(2^{\frac{17}{16}}\right)^8 = 2^{\frac{17}{16} \times 8} = 2^{\frac{136}{16}} = 2^{8.5}$$ 9. **Express the final answer:** $$2^{8.5} = 2^{8 + \frac{1}{2}} = 2^8 \times 2^{\frac{1}{2}} = 256 \times \sqrt{2}$$ **Final answer:** $$K = 256 \sqrt{2}$$