Question

What is `root4 (81x^8y^12)`?

Answer 1

`3x^2y^3`

Explanation:

`root4(81x^8y^12)`

Can be factored into four unit factors.

Each group of four can be taken out of the root leaving

`3x^2y^3`

Answer 2

`cancel(3x^2y^3)`
Better answer: As it is not defined if `y>0 " or "y<0`
Then we can have `+-3x^2y^3`

Explanation:

`x^8 -=( x^2)^4`

`y^12 -=(y^3)^4`

`3^4 -= 81`

Rewrite the given equation as

`root(4)(3^4(x^2)^4(y^3)^4) " " = " "3x^2y^3`

Answer 3

`3x^2abs(y^3)`

Explanation:

Note that `(3x^2y^3)^4 = 3^4(x^2)^4(y^3)^4 = 81x^8y^12`

So `3x^2y^3` is a fourth root of `81x^8y^12`, but it is not necessarily the one we want.

If an expression `E` is positive, then `root(4)(E)` denotes the positive fourth root of `E`.

If `x != 0` and `y < 0` then `y^3 < 0` and hence `3x^2y^3 < 0`, so this cannot be equal to the positive fourth root.

Note also that `(abs(y^3))^4 = y^12` and hence `(3x^2abs(y^3))^4 = 81x^8y^12` too.

Unlike `3x^2y^3`, the expression `3x^2abs(y^3)` is non-negative for all Real values of `x` and `y`. So `3x^2abs(y^3)` is the non-negative Real fourth root of `81x^8y^12` for all Real values of `x` and `y`.