Let P = [0, 4 ] : ^2 1 and S = a Z : 2( ^8 - ^8 ) 2 = a^2, P . Then n(S) is:

Step-by-step solution

Given the expression 2( ^8 - ^8 ) 2 = a^2 . We can simplify the term ^8 - ^8 as follows: ^8 - ^8 = ( ^4 - ^4 )( ^4 + ^4 ) ( ^2 - ^2 )( ^2 + ^2 )(( ^2 + ^2 )^2 - 2 ^2 ^2 ) 2 1 (1 - 1 2 ^2 2 ) Substitute this back into the given equation: 2 [ 2 (1 - 1 2 ^2 2 ) ] 1 2 = a^2 Since P , we have ^2 1 , which implies 2 0 . Thus, we can safely cancel 2 : 2 (1 - 1 2 ^2 2 ) = a^2 2 - ^2 2 = a^2 We know that for any real , 0 ^2 2 1 . However, since ^2 1 , ^2 2 1 . Therefore, 0 ^2 2 Multiplying by -1 and adding 2 , we get the ra