Find one solution for equation :$\displaystyle x^4 + y^\alpha = z^4 $
Where $\displaystyle x,y,z,\alpha $are positive integers.
"By inspection" means you take a look and try to guess values that will work. In this case I realized I could use $\displaystyle y^\alpha$ as a "wildcard", by setting alpha to 1, I could then set y to whatever I needed to make it work with the values I chose for x and z. Then I tried a few values of x and z that might work, by rewriting
$\displaystyle y^1 = z^4 - x^4$. Since the solution set is positive integers, z > x, so I chose z = 3, x = 2.