Solve y^12z^4-625

$$y ^ { 12 } z ^ { 4 } - 625$$

Factor

$$\left(zy^{3}-5\right)\left(zy^{3}+5\right)\left(z^{2}y^{6}+25\right)$$

Solution Steps

Rewrite y^{12}z^{4}-625 as \left(y^{6}z^{2}\right)^{2}-25^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(y^{6}z^{2}-25\right)\left(y^{6}z^{2}+25\right)$$

Reorder the terms.

$$\left(z^{2}y^{6}-25\right)\left(z^{2}y^{6}+25\right)$$

Consider z^{2}y^{6}-25. Rewrite z^{2}y^{6}-25 as \left(y^{3}z\right)^{2}-5^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(y^{3}z-5\right)\left(y^{3}z+5\right)$$

Reorder the terms.

$$\left(zy^{3}-5\right)\left(zy^{3}+5\right)$$

Rewrite the complete factored expression.

$$\left(zy^{3}-5\right)\left(zy^{3}+5\right)\left(z^{2}y^{6}+25\right)$$

Evaluate

$$z^{4}y^{12}-625$$