Let's use the fact that `4 = 2^2 ` and `8 = 2^3 . ` First, make the second expression base 2:

`4 ^ (8 ^ (2 ^ x)) = 2 ^ ( 2 * 8^ 2 ^ x ) ,`

so `4 ^ ( 8 ^ x ) = 2 *8 ^ ( 2 ^ x ) .`

These expressions can be also written base 2:

`2 ^ ( 2 * 8 ^x ) = 2 ^ ( 3 * 2 ^ x + 1 ) ,` so `2 * 8 ^x = 3 * 2 ^ x + 1 .`

Now denote `2^x = y , ` so we know `2 y^3 = 3 y + 1 . ` This equation has a root y = -1, so it can be divided by y + 1 using polynomial division. The result of this division is `2 y^2 - 2y = 1 .`

Finally, the expression in question is `y^2 - y , ` which is equal to 1/2.