Question

Q. x^-4/5=16 What is x Q. Suppose 0<a<1. Arrange the following from smallest to largest z^-a,z^a,z^1/a,z^-1/8.

Answer 1

x^-4/5=16
x^4/5=1/16
(x^4/5)^(5/4) = (1/16)^(5/4)
x = 1/32

your second is ambiguous

the terms are
(1/z)^a , z^a, z^(1/a) and (1/z)^(1/8)

it depends on the value of z
e.g. if z = 1, all terms have the same value
if z>1
investigate:
let z = 4, a = .2
(1/4)^.2 = .758
4^.2 = 1.32
4^(1/.2) = 1024
(1/4)^(1/8) = .84

arrange the terms according to the answers

now what about 0< z < 1
repeat my method and decide for that case

if z < 0 , that is if z is negative you will run into all kinds of undefined situations when you take roots.