what is in simplified radical form?

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embizze | High School Teacher | (Level 2) Educator Emeritus

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There are three rules for simplified radicals:

(1) There cannot be any fractions in the radicand. e.g. `sqrt(3/7)` is not allowed.

In order to fix this, use the property of radicals that `sqrt(a/b)=(sqrt(a))/(sqrt(b))`

So `sqrt(3/7)=sqrt(3)/sqrt(7)`

(2) There can be no radicals in the denominator. So `sqrt(3)/sqrt(7)` is not allowed.

In order to fix this you can rationalize: multiply the numerator and denominator by a radical so that the denominator is an integer.

e.g. `sqrt(3)/sqrt(7)*sqrt(7)/sqrt(7)=sqrt(21)/7` . Note that if the radical in the denominator has an index other than 2 (e.g. `root(3)(7)`) you rationalize by multiplying by the power that makes the product an integral power. For `root(3)(7)` you multiply by `root(3)(7^2)` since `7^(1/3)*7^(2/3)=7^1`

(3) There can be no perfect `n^(th)` powers in the radicand of index n. So no perfect square factors in a square root, no perfect cube factors in a cube root, etc...

In the case of a square root, factor out the largest square factor.

Ex: `sqrt(72)=sqrt(36*2)=sqrt(36)sqrt(2)=6sqrt(2)`

In the case of a cube root, factor out the largest cube factor.

Ex: `root(3)(16)=root(3)(8*2)=root(3)(8)root(3)(2)=2root(3)(2)`

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