Solve x^24-y^24

$$x ^ { 24 } - y ^ { 24 }$$

Factor

$$\left(x-y\right)\left(x+y\right)\left(x^{2}+y^{2}\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+xy+y^{2}\right)\left(x^{4}+y^{4}\right)\left(x^{4}-x^{2}y^{2}+y^{4}\right)\left(x^{8}-x^{4}y^{4}+y^{8}\right)$$

Solution Steps

Rewrite x^{24}-y^{24} as \left(x^{12}\right)^{2}-\left(y^{12}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{12}-y^{12}\right)\left(x^{12}+y^{12}\right)$$

Consider x^{12}-y^{12}. Rewrite x^{12}-y^{12} as \left(x^{6}\right)^{2}-\left(y^{6}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{6}-y^{6}\right)\left(x^{6}+y^{6}\right)$$

Consider x^{6}-y^{6}. Rewrite x^{6}-y^{6} as \left(x^{3}\right)^{2}-\left(y^{3}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{3}-y^{3}\right)\left(x^{3}+y^{3}\right)$$

Consider x^{3}-y^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).

$$\left(x-y\right)\left(x^{2}+xy+y^{2}\right)$$

Consider x^{3}+y^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x+y\right)\left(x^{2}-xy+y^{2}\right)$$

Consider x^{6}+y^{6}. Rewrite x^{6}+y^{6} as \left(x^{2}\right)^{3}+\left(y^{2}\right)^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x^{2}+y^{2}\right)\left(x^{4}-x^{2}y^{2}+y^{4}\right)$$

Consider x^{12}+y^{12}. Rewrite x^{12}+y^{12} as \left(x^{4}\right)^{3}+\left(y^{4}\right)^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x^{4}+y^{4}\right)\left(x^{8}-x^{4}y^{4}+y^{8}\right)$$

Rewrite the complete factored expression.

$$\left(x-y\right)\left(x+y\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+xy+y^{2}\right)\left(x^{4}-x^{2}y^{2}+y^{4}\right)\left(x^{8}-x^{4}y^{4}+y^{8}\right)\left(x^{2}+y^{2}\right)\left(x^{4}+y^{4}\right)$$

Evaluate

$$\left(x^{4}+y^{4}-\left(xy\right)^{2}\right)\left(-\left(xy\right)^{2}+\left(x^{2}+y^{2}\right)^{2}\right)\left(x^{8}-y^{8}\right)\left(x^{8}+y^{8}-\left(xy\right)^{4}\right)$$