(-8)^(1/3) would suffice.
I'm not sure exactly how to put that in your calculator though.
HP-15c power calculations from this owner's manual ...
http://hp15c.org/hp15c.pdf
(-8)^(1/3) is exactly what I am doing in my original post. If you look at the picture in skeeter's attachment it is the exact steps I am doing.
Input 8,
CHS (change sign) to make (-8),
Enter to store the (-8),
Then input 3,
1/x to get the reciprocal of 3 which is (1/3),
y^x to raise (-8) (which is y) to the (1/3) power (which is x).
I don't get what I am doing wrong, it should make sense enough to calculate...