Solve x^8-16y^8

$$x ^ { 8 } - 16 y ^ { 8 }$$

Factor

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

Solution Steps

Rewrite x^{8}-16y^{8} as \left(x^{4}\right)^{2}-\left(4y^{4}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

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

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

Rewrite the complete factored expression.

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

Evaluate

$$x^{8}-16y^{8}$$