The value of (2 12^ + 108^ )^2 + (2 12^ - 18^ )^2 is

Step-by-step solution

Let the given expression be E = (2 12^ + 108^ )^2 + (2 12^ - 18^ )^2 . Expanding both squares, we get: E = (4 ^2 12^ + ^2 108^ + 4 12^ 108^ ) + (4 ^2 12^ + ^2 18^ - 4 12^ 18^ ) Grouping the terms: E = 4( ^2 12^ + ^2 12^ ) + ( ^2 108^ + ^2 18^ ) + 4( 12^ 108^ - 12^ 18^ ) We know that 108^ = (90^ + 18^ ) = - 18^ . Therefore, ^2 108^ = ^2 18^ , and the second term becomes: ^2 18^ + ^2 18^ = 1 The first term is simply 4(1) = 4 . Now, simplify the cross terms: 4( 12^ (- 18^ ) - 12^ 18^ ) = -4( 18^ 12^ + 18^ 12^ ) Using