Have wasted the better part of day on this

This problem has been vexing me for a couple of days now. Can you get someone to point out the fallacy in reasoning?

(-8)^(-2/3)

If you input this into a calculator it displays a "Math Error"
If you google this it shows some imaginary value with a real part and an imaginary part as the answer

Here is how I (and almost everyone whom I asked) went about solving the problem:
(-8)^(-2/3)=((-2)^3)^(-2/3)=(-2)^(3*(-2/3)=(-2)^(-2)=(-1/2)^2=1/4

But if you raise both sides to the power of -3/2, the negative sign "magically" disappears:
((-8)^(-2/3))^(-3/2)=(1/4)^(-3/2)
OR
(-8)^((-2/3)*(-3/2))=((2)^(-2))^(-3/2))
OR
-8=(2)^((-2)*(-3/2))
OR
-8=2^3
OR
-8=8

One jugaad I applied here was to multiply the core on the RHS by (-1)^2
(1/4)^(-3/2)=(((-1)^2)*((2)^(-2)))^(-3/2)=((-1)^(2*(-3/2)))*((2)^((-2)*(-3/2)))=((-1)^(-3))*((2)^(3))=(-1)*8=-8

('a^b' implies 'a raised to the power of b'. * denotes multiplication, while 'a/b' represents a fraction, with 'a' as the numerator and 'b' as the denominator)

I now realize it has gotten a bit messy with all the brackets. Write it down on a piece of paper. Can you spot the fallacy in reasoning?

>>7
Why does the cube root of -1 equal (1 + i√3)/2 ?