With the expression #root(3)(81x^8y^12)#, we're taking the cube root of all the terms. We can write the cube root as a power of #1/3#, and so we get:
#(81 x^8 y^12)^(1/3)#
We can now distribute the #1/3# exponent to all the individual terms:
#81^(1/3) xx (x^8)^(1/3) xx (y^12)^(1/3)#
I'm going to express #81=3^4# and will also use the rule that #(x^a)^b=x^(ab)#:
#3^(4xx(1/3)) xx x^(8xx(1/3)) xx y^(12xx(1/3))#
which we can simplify:
#3^(4/3) xx x^(8/3) xx y^(12/3)#
#3^(4/3) xx x^(8/3) xx y^4#
We can now take the fractional exponents and where there is even divisibility, that lies outside of the root. Where there is a remainder, that can stay as a fraction or be shown under a root sign:
#3^(3/3) xx 3^(1/3) xx x^(6/3) xx x^(2/3) xx y^4#
#3x^2y^4 3^(1/3)x^(2/3)=3x^2y^4root(3)(3x^2)#
