1. **State the problem:** Find the value of $$\left((\log_2 16)^2\right)^{\frac{1}{\log_2(\log_2 16)}} \times \left(\sqrt{5}\right)^{\frac{1}{\log_5 5}}$$. 2. **Recall important rules:** - $$\log_a a = 1$$ for any positive $$a \neq 1$$. - $$a^{\log_a b} = b$$. - $$\sqrt{5} = 5^{\frac{1}{2}}$$. 3. **Evaluate inner logarithms:** - $$\log_2 16 = 4$$ because $$2^4 = 16$$. - Then, $$\log_2(\log_2 16) = \log_2 4 = 2$$ because $$2^2 = 4$$. - Also, $$\log_5 5 = 1$$. 4. **Rewrite the expression with these values:** $$\left(4^2\right)^{\frac{1}{2}} \times \left(5^{\frac{1}{2}}\right)^1$$ 5. **Simplify powers:** - $$4^2 = 16$$ - So, $$\left(16\right)^{\frac{1}{2}} = \sqrt{16} = 4$$ - And $$\left(5^{\frac{1}{2}}\right)^1 = 5^{\frac{1}{2}} = \sqrt{5}$$ 6. **Multiply the results:** $$4 \times \sqrt{5}$$ 7. **Evaluate $$\sqrt{5}$$ approximately:** $$\sqrt{5} \approx 2.236$$ 8. **Final value:** $$4 \times 2.236 = 8.944$$ which is approximately 9, not exactly matching any options. 9. **Check if simplification can be exact:** Note that the problem likely expects simplification without decimal approximation. 10. **Re-express the original expression carefully:** $$\left((\log_2 16)^2\right)^{\frac{1}{\log_2(\log_2 16)}} = \left(4^2\right)^{\frac{1}{2}} = 16^{\frac{1}{2}} = 4$$ $$\left(\sqrt{5}\right)^{\frac{1}{\log_5 5}} = \left(5^{\frac{1}{2}}\right)^1 = 5^{\frac{1}{2}} = \sqrt{5}$$ So the product is $$4 \times \sqrt{5}$$. 11. **Check if $$4 \times \sqrt{5}$$ equals any of the options:** - (A) 18 - (B) 16 - (C) 8 Since $$4 \times \sqrt{5} \approx 8.944$$, closest to 8. 12. **Conclusion:** The best matching answer is (C) 8. **Final answer:** (C) 8