Rationalize: 1/³√x + ³√y

show step by step solution to explain the answer

### 2 Answers | Add Yours

`1/root(3)x + root(3)y`

`= (1+root(3)(x)xxroot(3)y)/root(3)x`

Multiplying both numerator and denominator by `(root(3)x)^2`

`=((root(3)x)^2(1+root(3)(x)xxroot(3)y))/((root(3)x)^2root(3)x)`

`= ((root(3)x)^2+xroot(3)y)/x`

Therefore,

`1/root(3)x+root(3)y = ((root(3)x)^2+xroot(3)y)/x`

or if you asked to rationalize `1/(root(3)x+root(3)y)`

We know,

`(x+y) = ((root(3)x)^3+(root(3)y)^3)`

`((root(3)x)^3+(root(3)y)^3) = (root(3)x)+root(3)y)((root(3)x)^2-root(3)xroot(3)y+(root(3)y)^2)`

Therefore,

`1/(root(3)x+root(3)y) = ((root(3)x)^2-root(3)xroot(3)y+(root(3)y)^2)/(x+y)`

**This is the rationalized form.**

### Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes