Question

How do you simplify `sqrt(108x^5y^6)`?

Answer 1

If you only want integer powers, then the following is simplified:

`=>6y^3sqrt(3x^5)`

If you are okay with fractional powers, then you could equivalently write:

`=>(6sqrt(3))x^(5/2)y^3`

Explanation:

We are given:

`sqrt(108x^5y^6)`

First, let's deal with the constant term `108`. We look for factors of `108` that are perfect squares, as this will allow us to pull it out of the radical.

Factors of `108 = {color(blue)1,2,3,color(blue)4,6,color(blue)9,12,18,27,color(blue)36,54,108}`.

The factors in `color(blue)"blue"` are perfect squares. To simplify the most, we want the largest one. So we choose `36`, which multiplied by `3` gives `108`.

`sqrt((36)(3) x^5y^6)`

`=> sqrt((6^2)(3)x^5y^6)`

`=> 6sqrt(3x^5y^6)`

We can also simplify the variable powers. Taking the square root is equivalent to raising to the `1/2` power. If you only want integer powers, then the following is simplified:

`6y^(6/2)sqrt(3x^5)`

`=>6y^3sqrt(3x^5)`

If you are okay with fractional powers, then you could equivalently write:

`=>(6sqrt(3))x^(5/2)y^3`