  We study equations in groups G with unique m-th roots for each positive
integer m. A word equation in two letters is an expression of the form w(X,A) =
B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as
fixed coefficients, and X in G as the unknown. Certain word equations, such as
XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do
not. We obtain the first known infinite families of word equations not solvable
by radicals, and conjecture a complete classification. To a word w we associate
a polynomial P_w in Z[x,y] in two commuting variables, which factors whenever w
is a composition of smaller words. We prove that if P_w(x^2,y^2) has an
absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not
solvable in terms of radicals.
