Question 148127
{{{root(3,40a^3b^6c^8)}}} Start with the given expression.



{{{root(3,8*5a^3b^6c^8)}}}  Factor {{{40}}} into {{{8*5}}}. Notice how 8 is a perfect cube (ie {{{8=2^3}}}). 



{{{root(3,8*5a^3b^3*b^3c^8)}}} Factor {{{b^6}}} into {{{b^3*b^3}}}. Once again, notice how {{{b^3}}} is a perfect cube. 


{{{root(3,8*5a^3b^3*b^3c^8)}}} Factor {{{c^8}}} into {{{c^3*c^3*c^2}}}. Once again, notice how {{{c^3}}} is a perfect cube. 



I'm breaking the expression into perfect cubes so that when I take the cube root of these cubes, I'll be left with the expression itself. Since {{{b^3}}} is a perfect cube, this means that {{{root(3,b^3)=b}}}. Also, since {{{c^3}}} is a perfect cube, this means that {{{root(3,c^3)=c}}}.



{{{root(3,8)*root(3,5)*root(3,a^3)*root(3,b^3)*root(3,b^3)*root(3,c^3)*root(3,c^3)*root(3,c^2)}}} Break up the root.



{{{2*root(3,5)*a*b*b*c*c*root(3,c^2)}}} Take the cube root of the perfect cubes to get just the base of the expression.


{{{2ab^2c^2*root(3,5c^2)}}} Recombine any roots leftover and multiply



So {{{root(3,40a^3b^6c^8)}}} simplifies to {{{2ab^2c^2*root(3,5c^2)}}}



In other words, {{{root(3,40a^3b^6c^8)=2ab^2c^2*root(3,5c^2)}}}.