If
r
,
s
r,s
are nonzero integers and
m
m
is the largest squarefree divisor of
r
s
rs
, then for every element
z
z
in the alternating group
A
n
{A_n}
, the equation
z
=
x
r
y
s
z = {x^r}{y^s}
has a solution with
x
,
y
∈
A
n
x,y \in {A_n}
, provided that
n
⩾
5
n \geqslant 5
and
n
⩾
(
5
/
2
)
log
m
n \geqslant (5/2)\log m
. The bound
(
5
/
2
)
log
m
(5/2)\log m
improves the bound
4
m
+
1
4m + 1
of Droste. If
n
⩾
29
n \geqslant 29
, the coefficient
5
/
2
5/2
may be replaced by 2; however,
5
/
2
5/2
cannot be replaced by 1 even for all large
n
n
.