Solve x^6-4^6

$$x ^ { 6 } - 4 ^ { 6 }$$

Factor

$$\left(x-4\right)\left(x+4\right)\left(x^{2}-4x+16\right)\left(x^{2}+4x+16\right)$$

Solution Steps

Rewrite x^{6}-4096 as \left(x^{3}\right)^{2}-64^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x^{3}-64\right)\left(x^{3}+64\right)$$

Consider x^{3}-64. Rewrite x^{3}-64 as x^{3}-4^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).

$$\left(x-4\right)\left(x^{2}+4x+16\right)$$

Consider x^{3}+64. Rewrite x^{3}+64 as x^{3}+4^{3}. The sum of cubes can be factored using the rule: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right).

$$\left(x+4\right)\left(x^{2}-4x+16\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: x^{2}-4x+16,x^{2}+4x+16.

$$\left(x-4\right)\left(x+4\right)\left(x^{2}-4x+16\right)\left(x^{2}+4x+16\right)$$

Evaluate

$$\left(x^{2}-16\right)\left(\left(x^{2}+16\right)^{2}-16x^{2}\right)$$

Solution Steps

Calculate 4 to the power of 6 and get 4096.

$$x^{6}-4096$$