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 :

            $(2x^8 - 3y^2)*(4x^{16} + 6x^8y^2 + 9y^4)$