General rules we need to know:
[1]#color(white)("XXXX")b^(-p) = 1/(b^p)#
[2]#color(white)("XXXX")b^(q/r) = (b^(1/r))^q#
[3]#color(white)("XXXX")b^(1/r) = root(r)(b)#
[4]#color(white)("XXXX")root(r)(b/a) = root(r)(b)/root(r)(a)#
Given the above:
#color(white)("XXXXXXXX")(-27/8)^(-4/3) = (-8/27)^(4/3color(white)("XXXX")#Rule [1]
#color(white)("XXXX")(-8/27)^(4/3) =( (-8/27)^(1/3))^4color(white)("XXXXX")#Rule [2]
#color(white)("XXXX")( (-8/27)^(1/3))^4 = (-root(3)(8)/root(3)(27))^4color(white)("XXXX")#Rules [3] and [4]
#color(white)("XXXX")(-root(3)(8)/root(3)(27))^4 = (-2/3)^4color(white)("XXXXXXX")#Root extraction by observation
#color(white)("XXXX")(-2/3)^4 = 16/81color(white)("XXXXXXXXXXXXX")#Repeated multiplication