Solve s^8-6561

$$s ^ { 8 } - 6561$$

Factor

$$\left(s-3\right)\left(s+3\right)\left(s^{2}+9\right)\left(s^{4}+81\right)$$

Solution Steps

Rewrite s^{8}-6561 as \left(s^{4}\right)^{2}-81^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(s^{4}-81\right)\left(s^{4}+81\right)$$

Consider s^{4}-81. Rewrite s^{4}-81 as \left(s^{2}\right)^{2}-9^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(s^{2}-9\right)\left(s^{2}+9\right)$$

Consider s^{2}-9. Rewrite s^{2}-9 as s^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(s-3\right)\left(s+3\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: s^{2}+9,s^{4}+81.

$$\left(s-3\right)\left(s+3\right)\left(s^{2}+9\right)\left(s^{4}+81\right)$$

Evaluate

$$s^{8}-6561$$