Answer:
[tex](x^{4}y^{6}+1)(x^{8}y^{12}-x^{4}y^{6}+1)[/tex].
Step-by-step explanation:
We want to expand: [tex]x^{12}y^{18}+1[/tex].
We can rewrite this as the sum of two cubes.
[tex](x^{4})^3(y^{6})^3+1=(x^{4}y^{6})^3+1^3[/tex].
Recall and use the sum of cubes identity: [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]
By comparing our newly rewritten expression to this identity, we have [tex]a=x^4y^6[/tex] and [tex]b=1[/tex].
We substitute into the identity to get:
[tex](x^{4}y^{6})^3+1^3=[x^{4}y^{6}+1][(x^{4}y^{6})^2-(x^{4}y^{6})(1)+1^2][/tex].
We now use this rule of exponents :[tex](a^m)^n=a^{mn}[/tex] to get;
[tex](x^{4}y^{6}+1)(x^{8}y^{12}-x^{4}y^{6}+1)[/tex].
