Solve a^6-6^6=

$$a ^ { 6 } - 6 ^ { 6 } =$$

Factor

$$\left(a-6\right)\left(a+6\right)\left(a^{2}-6a+36\right)\left(a^{2}+6a+36\right)$$

Solution Steps

Rewrite a^{6}-46656 as \left(a^{3}\right)^{2}-216^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a^{3}-216\right)\left(a^{3}+216\right)$$

Consider a^{3}-216. Rewrite a^{3}-216 as a^{3}-6^{3}. The difference of cubes can be factored using the rule: p^{3}-q^{3}=\left(p-q\right)\left(p^{2}+pq+q^{2}\right).

$$\left(a-6\right)\left(a^{2}+6a+36\right)$$

Consider a^{3}+216. Rewrite a^{3}+216 as a^{3}+6^{3}. The sum of cubes can be factored using the rule: p^{3}+q^{3}=\left(p+q\right)\left(p^{2}-pq+q^{2}\right).

$$\left(a+6\right)\left(a^{2}-6a+36\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: a^{2}-6a+36,a^{2}+6a+36.

$$\left(a-6\right)\left(a+6\right)\left(a^{2}-6a+36\right)\left(a^{2}+6a+36\right)$$

Evaluate

$$\left(a^{2}-36\right)\left(\left(a^{2}+36\right)^{2}-36a^{2}\right)$$

Solution Steps

Calculate 6 to the power of 6 and get 46656.

$$a^{6}-46656$$