1. The problem is to simplify the expression $((2 \times 2)^2)^2)^2)^2)^2)^2$. 2. First, simplify inside the parentheses: $2 \times 2 = 4$. So the expression becomes $((4)^2)^2)^2)^2)^2)^2$. 3. Apply the exponentiation step by step from the innermost parentheses outward. Recall the rule $(a^m)^n = a^{m \times n}$. 4. Start with $(4)^2 = 4^2 = 16$. So the expression is now $((16)^2)^2)^2)^2)^2$. 5. Next, $(16)^2 = 16^2 = 256$. Expression becomes $((256)^2)^2)^2)^2$. 6. Then, $(256)^2 = 256^2 = 65536$. Expression becomes $((65536)^2)^2)^2$. 7. Next, $(65536)^2 = 65536^2 = 4294967296$. Expression becomes $((4294967296)^2)^2$. 8. Then, $(4294967296)^2 = 4294967296^2 = 18446744073709551616$. Expression becomes $(18446744073709551616)^2$. 9. Finally, $(18446744073709551616)^2 = 18446744073709551616^2 = 340282366920938463463374607431768211456$. 10. Therefore, the simplified value of the original expression is $340282366920938463463374607431768211456$.