Question

How do you simplify `(-91a^4b+14ab )/ (-7ab)`?

Answer 1

`(-91a^4b+14ab)/(-7ab)`

`= 13a^3-2`

`=(root(3)(13)a-root(3)(2))(root(3)(169)a^2+root(3)(26)a+root(3)(4))`

with exclusions `a!=0` and `b!=0`

Explanation:

Both of the terms `-91a^4b` and `14ab` are divisible by `-7ab`, so we find:

`(-91a^4b+14ab)/(-7ab) = 13a^3-2`

with exclusions `a!=0` and `b!=0`

Note that the exclusions are required because if `a=0` or `b=0` then the original expression is of the form `0/0` which is undefined.

I guess that counts as 'simplified', but it can be factored too, using cube roots...

For any Real numbers `x` and `y` note that:

`root(3)(x) root(3)(y) = root(3)(xy)`

We will use this below.

The difference of cubes identity can be written:

`A^3-B^3 = (A-B)(A^2+AB+B^2)`

We can factor `13a^3-2` using this identity with `A=root(3)(13)a` and `b = root(3)(2)` as follows:

`13a^3-2`

`=(root(3)(13)a)^3-(root(3)(2))^3`

`=(root(3)(13)a-root(3)(2))((root(3)(13)a)^2+(root(3)(13)a)(root(3)(2))+(root(3)(2))^2)`

`=(root(3)(13)a-root(3)(2))(root(3)(169)a^2+root(3)(26)a+root(3)(4))`