Question

How do you simplify `(-2x+root3(-5x^4y^3))/(3root3(15x^3y))`?

Answer 1

`(-2 -yroot(3)(5x))/(3root(3)(15y))`

Explanation:

First, pull any cubes out of the cube roots. We know that:

`root(3)(color(red)a^3) = color(red)a`

And therefore,

`root(3)(color(red)a^3 xx b xx c xx cdots) = color(red)a * root(3)(b xx c xx cdots`

We can use this to pull stuff out of the cube roots which doesn't need to be in there. So let's do that `-` I'll highlight any cubes with different colors so it's clearer what I'm doing when I change things around.

`(-2x + root(3)(-5x^4y^3))/(3root(3)(15x^3y))`

`= (-2x + root(3)(5(-1)(x^3)(x)(y^3)))/(3root(3)(15(x^3)(y)))`

`= (-2x + root(3)(5(color(red)(-1))^3(color(blue)x^3)(x)(color(orange)y^3)))/(3root(3)(15(color(limegreen)x^3)(y)))`

`= (-2x + (color(red)(-1))(color(blue)x)(color(orange)y) root(3)(5x))/(3(color(limegreen)x)root(3)(15(y)))`

`= (-2x -xyroot(3)(5x))/(3xroot(3)(15y))`

At this point, we can cancel out an `x` on the top and the bottom. Notice that all terms have an `x` outside the radical, so this is possible.

`= ((x)(-2 -yroot(3)(5x)))/((x)(3root(3)(15y)))`

`= (cancel((x))(-2 -yroot(3)(5x)))/(cancel((x))(3root(3)(15y)))`

`= (-2 -yroot(3)(5x))/(3root(3)(15y))`

We could technically simplify this further by rationalizing the denominator, but this is a pretty good stopping point as far as simplification goes.

Final Answer