Question 4028
 It is very good that you typed the expression corectly and clearly .
 On way
 (x+1)^(-3/2)=8
 [(x+1)^(-1/2)]3 = 2^3,
 So, (x+1)^(-1/2) = 2.
 Square on both sides:
 (x+1)^(-1) =  4,
 Or 1/(x+1) = 4,
 So, (x+1) = 1/4,
 x = -3/4.

 Another way:
 Square on both sides:
 (x+1)^(-3) = 8^2 = 64 = 2^6,
 Take cubic root on both sides:
 (x+1)^(-1) = (2^6)^(1/3) = 2^2 =4,
 So, x+1 = 1/4, x = -3/4

 Check  (-3/4 + 1)^(-3/2) = (1/4)^(-3/2) = (1/2)^2*(-3/2)
       = (1/2)^(-3) = [(1/2)^(-1)]^3 
       = 2^3 = 8 (correct)

 Kenny