  It is known that the fifth Engel word $E_5$ is trivial in the 2-generator
group of exponent four $B(2,4)$, and so can be written as a product of fourth
powers. Explicit products of 250 and 28 powers are known, using fourth powers
of words up to lengths four and ten respectively. Using a reduction technique
based on the recursive enumerability of the set of trivial words in a finite
presentation we were able to rewrite $E_5$ as a product of 26 fourth powers of
words up to length five.
