Calculate. 1 ( x ( 1 / 3 ) + y ( 1 / 3 ) )

Question

Calculate.  1     (      x          (      1              /            3      )        +      y          (      1              /            3      )        )

Cael Cox · Accepted Answer

1      /    (      x          (      1              /            3      )        +      y          (      1              /            3      )        )  =  (      x          (      2              /            3      )        +      y          (      2              /            3      )        −  (  x  y      )          (      1              /            3      )        )      /    (  x  +  y  )ie. multiply and divide by    (      x          (      2              /            3      )        +      y          (      2              /            3      )        −  (  x  y      )          (      1              /            3      )        ).

Nash Frank · Accepted Answer

I assume you are required to rationalize the denominator.
Use the sum of cubes formula,

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

for your question use

x + y = (x^{1 / 3} + y^{1 / 3}) (x^{2 / 3} - x^{1 / 3} y^{1 / 3} + y^{2 / 3})

multiply numerator and denominator by

(x^{2 / 3} - x^{1 / 3} y^{1 / 3} + y^{2 / 3})

to get

(x^{2 / 3} - x^{1 / 3} y^{1 / 3} + y^{2 / 3}) / (x + y)

Calculate. 1\ (x^(1/3)+y^(1/3))

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