Question:
Find the value
$x^{6}+y^{6}$
Solution:
$=\left(x^{2}\right)^{3}+\left(y^{2}\right)^{3}$
$=\left(x^{2}+y^{2}\right)\left(\left(x^{2}\right)^{2}-x^{2} y^{2}+\left(y^{2}\right)^{2}\right)$
$=\left(x^{2}+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)$
$\left[\therefore a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$
$\therefore x^{6}+y^{6}=\left(x^{2}+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)$