Solve a^8-6^8

$$a ^ { 8 } - 6 ^ { 8 }$$

Factor

$$\left(a-6\right)\left(a+6\right)\left(a^{2}+36\right)\left(a^{4}+1296\right)$$

Solution Steps

Rewrite a^{8}-1679616 as \left(a^{4}\right)^{2}-1296^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a^{4}-1296\right)\left(a^{4}+1296\right)$$

Consider a^{4}-1296. Rewrite a^{4}-1296 as \left(a^{2}\right)^{2}-36^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

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

Consider a^{2}-36. Rewrite a^{2}-36 as a^{2}-6^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a-6\right)\left(a+6\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: a^{2}+36,a^{4}+1296.

$$\left(a-6\right)\left(a+6\right)\left(a^{2}+36\right)\left(a^{4}+1296\right)$$

Evaluate

$$a^{8}-1679616$$

Solution Steps

Calculate 6 to the power of 8 and get 1679616.

$$a^{8}-1679616$$