How do you use the binomial theorem to expand #(x^(2/3)-y^(1/3))^3#?

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Answer 1 · 2017-07-27T09:11:41

The answer is #=x^2-3x^(4/3)y^(1/3)+3x^(2/3)y^(2/3)-y#

We need

#(a-b)^3=((3),(0))a^3-((3),(1))a^2b+((3),(2))ab^2-((3),(3))b^3#

#(a-b)^3=a^3-3a^2b+3ab^2-b^3#

In our case,

#a=x^(2/3)#

and #b=y^(1/3)#

Therefore,

#(x^(2/3)-y^(1/3))=( x^(2/3 ))^ 3-3*(x^(2/3))^2*y^(1/3) +3*x^(2/3)*(y^(1/3))^2-(y^(1/3))^3#

#=x^2-3x^(4/3)y^(1/3)+3x^(2/3)y^(2/3)-y#