Question

What is the factored form of 8x^24-27y^6

Answer 1

Factoring:
`8x^(24)-27y^6`

Theory : A difference of two perfect cubes,
`a^3-b^3` can be factored into

`(a-b)*(a^2+ab+b^2)`

Proof :
`(a-b)*(a^2+ab+b^2)=`

`a^3+a^2b+ab^2-ba^2-b^2a-b^3=`

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

`a^3+0+0+b^3=`

`a^3+b^3`

Check : 8 is the cube of 2

Check : 27 is the cube of 3

Check :
`x^(24)` is the cube of
`x^8`

Check :
`y^6` is the cube of
`y^2`

Factorization is :

`(2x^8 - 3y^2)*(4x^(16) + 6x^8y^2 + 9y^4)`