Question 200828
{{{3*root(3,y^5)+4*root(3,y^2)+root(3,8y^6)}}} Start with the given expression.



{{{3*root(3,y^5)+4*root(3,y^2)+root(3,2^3*y^6)}}} Rewrite 8 as {{{2^3}}}



{{{3*root(3,y^3*y^2)+4*root(3,y^2)+root(3,2^3*y^6)}}} Rewrite {{{y^5}}} as {{{y^3*y^2}}}



{{{3*root(3,y^3*y^2)+4*root(3,y^2)+root(3,2^3*y^3*y^3)}}} Rewrite {{{y^6}}} as {{{y^3*y^3}}}



{{{3*root(3,y^3)*root(3,y^2)+4*root(3,y^2)+root(3,2^3)*root(3,y^3)*root(3,y^3)}}} Break up the roots using the identity {{{root(n,x*y)=root(n,x)*root(n,y)}}}



{{{3*root(3,y^3)*root(3,y^2)+4*root(3,y^2)+2*root(3,y^3)*root(3,y^3)}}} Evaluate the cube root of {{{2^3}}} to get 2



{{{3*y*root(3,y^2)+4*root(3,y^2)+2*y*y}}} Evaluate the cube root of {{{y^3}}} to get "y"



{{{3*y*root(3,y^2)+4*root(3,y^2)+2y^2}}} Multiply



{{{(3y+4)*root(3,y^2)+2y^2}}} Factor out the {{{root(3,y^2)}}} from the first two terms. note: this step isn't necessary, but your book does it.



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Answer:



So {{{3*root(3,y^5)+4*root(3,y^2)+root(3,8y^6)}}} simplifies to {{{(3y+4)*root(3,y^2)+2y^2}}}




In other words, {{{3*root(3,y^5)+4*root(3,y^2)+root(3,8y^6)=(3y+4)*root(3,y^2)+2y^2}}}