Question

How do you factor completely `8x^6n - 27y^9m`?

Answer 1

If `m, n` are Real constants, then:

`8x^6n-27y^9m`

`=(2root(3)(n)x^2-3root(3)(m)y^3)(4root(3)(n^2)x^4+6root(3)(mn)x^2y^3+9root(3)(m^2)y^6)`

Explanation:

If we assume that `n, m` are Real constants and `x, y` are variables, then it is possible to factor this expression as a difference of cubes.

The difference of cubes identity can be written:

`a^3-b^3 = (a-b)(a^2+ab+b^2)`

So we find:

`8x^6n-27y^9m`

`=(2root(3)(n)x^2)^3-(3root(3)(m)y^3)^3`

`=(2root(3)(n)x^2-3root(3)(m)y^3)((2root(3)(n)x^2)^2+(2root(3)(n)x^2)(3root(3)(m)y^3)+(3root(3)(m)y^3)^2)`

`=(2root(3)(n)x^2-3root(3)(m)y^3)(4root(3)(n^2)x^4+6root(3)(mn)x^2y^3+9root(3)(m^2)y^6)`