One of the laws of indices states: x^(m/n) = rootn((x^m)) = (rootn(x))^mxmn=n√(xm)=(n√x)m
Therefore 8^(2/3)823 can be written as root3((8^2)) or (root3(8))^23√(82)or(3√8)2
I prefer to use the second form because it uses smaller numbers - the root is found first and then that is squared.
root3(8) = 2 and 2^2 =43√8=2and22=4
So: (color(blue)(root3(8)))^2 = color(blue)(2)^2 = 4(3√8)2=22=4
Consider a question such as 32^(3/5)3235
root5(color(blue)((32^3)))5√(323) would mean finding 32^3323 first ... ouch!
(color(blue)(root5 (32)))^3(5√32)3 would mean finding root5(32)5√32 first. That is color(blue)(2)2
(color(blue)(root5 (32)))^3 = color(blue)(2)^3 = 8(5√32)3=23=8
