Solve 625m^8n^4-16p^8

$$625 m ^ { 8 } n ^ { 4 } - 16 p ^ { 8 }$$

Factor

$$\left(5nm^{2}-2p^{2}\right)\left(5nm^{2}+2p^{2}\right)\left(25n^{2}m^{4}+4p^{4}\right)$$

Solution Steps

Rewrite 625m^{8}n^{4}-16p^{8} as \left(25m^{4}n^{2}\right)^{2}-\left(4p^{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).

$$\left(25m^{4}n^{2}-4p^{4}\right)\left(25m^{4}n^{2}+4p^{4}\right)$$

Reorder the terms.

$$\left(25n^{2}m^{4}-4p^{4}\right)\left(25n^{2}m^{4}+4p^{4}\right)$$

Consider 25n^{2}m^{4}-4p^{4}. Rewrite 25n^{2}m^{4}-4p^{4} as \left(5m^{2}n\right)^{2}-\left(2p^{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).

$$\left(5m^{2}n-2p^{2}\right)\left(5m^{2}n+2p^{2}\right)$$

Reorder the terms.

$$\left(5nm^{2}-2p^{2}\right)\left(5nm^{2}+2p^{2}\right)$$

Rewrite the complete factored expression.

$$\left(5nm^{2}-2p^{2}\right)\left(5nm^{2}+2p^{2}\right)\left(25n^{2}m^{4}+4p^{4}\right)$$

Evaluate

$$625n^{4}m^{8}-16p^{8}$$