Simplify ^3 the square root of 21 divided by ^3 the square root of 7xy^2.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Simplify `(root(3)(21))/(root(3)(7xy^2))` :

Radical expressions are simplified if there are no radicals in the denominator, no fractions in the radicand, and no perfect nth power facors of the radicand (where n is the index.)

We can get rid of the radical in the denominator by "rationalizing"; multiply numerator and denominator by an expression so that the radicand in the denominator is a perfect cube.

`(root(3)(21))/(root(3)(7xy^2))*(root(3)(49x^2y))/(root(3)(49x^2y))`

`=(root(3)(1029x^2y))/(root(3)(343x^3y^3))`

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

`=(root(3)(3x^2y))/(xy)` which is simplified.

Note that we could have simplified a bit at the beginning using `(root(n)(a))/(root(n)(b))=root(n)(a/b)` to get:

`(root(3)(3))/(root(3)(xy^2))`

then `(root(3)(3))/(root(3)(x^2y))*(root(3)(x^2y))/(root(3)(x^2y))`

`=(root(3)(3x^2y))/(root(3)(x^3y^3))=(root(3)(3x^2y))/xy` as before.

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