Factorise the following: $x^{8} - y^{8}$

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Solution

Given: $x^{8} - y^{8}$
$= (x^{4})^{2} - (y^{4})^{2}$ [ $a^{2} - b^{2} = (a + b) (a - b)$ ]
$= (x^{4} + y^{4}) (x^{4} - y^{4})$ [ $a^{2} - b^{2} = (a + b) (a - b)$ ]
$= (x^{4} + y^{4}) [(x^{2})^{2} - (y^{2})^{2}]$ [ $a^{2} - b^{2} = (a + b) (a - b)$ ]
$= (x^{4} + y^{4}) (x^{2} + y^{2}) (x^{2} - y^{2})$ [ $a^{2} - b^{2} = (a + b) (a - b)$ ]
$= (x^{4} + y^{4}) (x^{2} + y^{2}) (x + y) (x - y)$

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