Question

Question #79897

Answer 1

`root(3)(7x)/root(3)(3y)=root(3)(63xy^2)/(3y)`

Explanation:

Assuming "rationalize" refers to rationalizing the denominator , note that `root(3)(3y)` is the value such that `(root(3)(3y))^3 = 3y`. We will use that fact to eliminate the root from the denominator.

`root(3)(7x)/root(3)(3y) = (root(3)(7x) * (root(3)(3y))^2) / (root(3)(3y) * (root(3)(3y))^2)`

`=(root(3)(7x) * root(3)((3y)^2))/(root(3)(3y))^3`

`=root(3)(7x*(3y)^2)/(3y)`

`=root(3)(63xy^2)/(3y)`

(Note that we also used the properties `(root(a)(x))^b = root(a)(x^b)` and `root(a)(x)*root(a)(y) = root(a)(xy)` during the process)