Solve x^36-16

$$x ^ { 36 } - 16$$

Factor

$$\left(x^{9}-2\right)\left(x^{9}+2\right)\left(x^{18}+4\right)$$

Solution Steps

Rewrite x^{36}-16 as \left(x^{18}\right)^{2}-4^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{18}-4\right)\left(x^{18}+4\right)$$

Consider x^{18}-4. Rewrite x^{18}-4 as \left(x^{9}\right)^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{9}-2,x^{9}+2,x^{18}+4.

$$\left(x^{9}-2\right)\left(x^{9}+2\right)\left(x^{18}+4\right)$$

Evaluate

$$x^{36}-16$$