Solve a^49+b^7

$${ a }^{ 49 } + { b }^{ 7 }$$

Factor

$$\left(a^{7}+b\right)\left(a^{42}+b^{6}-b^{5}a^{7}+b^{4}a^{14}-b^{3}a^{21}+b^{2}a^{28}-ba^{35}\right)$$

Solution Steps

Consider a^{49}+b^{7} as a polynomial over variable a. Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{49} and m divides the constant factor b^{7}. One such factor is a^{7}+b. Factor the polynomial by dividing it by this factor.

$$\left(a^{7}+b\right)\left(a^{42}+b^{6}-b^{5}a^{7}+b^{4}a^{14}-b^{3}a^{21}+b^{2}a^{28}-ba^{35}\right)$$

Evaluate

$$a^{49}+b^{7}$$