I have two elements of the form $$ w = x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} x_6^{a_6} $$ and $$ w' = x_1^{b_1} x_2^{b_2} x_3^{b_3} x_4^{b_4} x_5^{b_5} x_6^{b_6} $$ for integers $a_i$ and $b_j$ where the $a_i$ are the same as the $b_j$ as multisets and I'd like to know if there is always an automorphism of $F_6$ sending $w$ to $w'$. Is this the case? For example, if in $F_2$ I knew that $x_1^{a_1}x_2^{a_2}$ and $x_2^{a_2}x_1^{a_1}$ were always related by an automorphism, I'd be all set.
Edit: In my original question I asked for about the existence of an algorithm to tell if two words in a free group differ by applying an automorphism. This is completely answered by Whitehead's algorithm, as Derek Holt pointed out. I'm still interested in the question regarding the specific elements that I list here, hence the edit. To be even more concrete, how about the elements $$ w = x_1^2 x_2^2 x_3^3 x_4^3 x_5^5 x_6^5 $$ and $$ w' = x_1^3 x_2^2 x_3^5 x_4^3 x_5^5 x_6^2 $$ in $F_6$.