1. **State the problem:** Simplify the expression $$\frac{\sqrt{4^{2n} \cdot 4} \times 8^{5-2n}}{\sqrt{16^{6-2n}}}$$. 2. **Rewrite bases as powers of 2:** - $4 = 2^2$ - $8 = 2^3$ - $16 = 2^4$ 3. **Rewrite each term:** - $4^{2n} = (2^2)^{2n} = 2^{4n}$ - $4 = 2^2$ - So, $4^{2n} \cdot 4 = 2^{4n} \cdot 2^2 = 2^{4n+2}$ - $8^{5-2n} = (2^3)^{5-2n} = 2^{15 - 6n}$ - $16^{6-2n} = (2^4)^{6-2n} = 2^{24 - 8n}$ 4. **Rewrite the expression using these:** $$\frac{\sqrt{2^{4n+2}} \times 2^{15 - 6n}}{\sqrt{2^{24 - 8n}}}$$ 5. **Simplify the square roots:** - $\sqrt{2^{4n+2}} = 2^{\frac{4n+2}{2}} = 2^{2n+1}$ - $\sqrt{2^{24 - 8n}} = 2^{\frac{24 - 8n}{2}} = 2^{12 - 4n}$ 6. **Substitute back:** $$\frac{2^{2n+1} \times 2^{15 - 6n}}{2^{12 - 4n}}$$ 7. **Combine numerator powers:** $$2^{2n+1 + 15 - 6n} = 2^{16 - 4n}$$ 8. **Divide powers by subtracting exponents:** $$2^{16 - 4n} \div 2^{12 - 4n} = 2^{(16 - 4n) - (12 - 4n)} = 2^{16 - 4n - 12 + 4n} = 2^{4}$$ 9. **Final answer:** $$2^{4} = 16$$ Thus, the simplified value of the expression is **16**.