How do you know when a radical is in simplest form?

My answer: All and any perfect squares under the radical were identified, the sqrt has been taken out of each perfect square, and each root has been taken out from under the sqrt sign and written in front instead.
Is this correct?

1 Answer
Feb 24, 2017

Answer:

See explanation

Explanation:

Prime factors are the key

Considering just numbers.

Suppose we had had a square root and the number can be written as a product of primes. For example: #sqrt(2^2 xx3^3xx3)#

Then we have #2xx3xxsqrt(3)=6sqrt(3)#

The 3 inside the root can not be broken down any further into whole number factors. So that is the point you stop.

If you have a variable, for example #sqrt(2^2xx2x)#

We can take the #2^2# out of the root but we do not know the value of #x#. Consequently we can not factor out any prime number elements of it.

However; if you had #sqrt(2^2xx2xx x^2)# then you could take the #x# outside the root giving: #2xsqrt(2)#