Given a field
F
F
, a positive integer
m
m
and an integer
n
≥
2
n\geq 2
, it is proved that the symbol length of classes in Milnor’s
K
K
-groups
K
n
F
/
2
m
K
n
F
K_n F/2^m K_n F
that are equivalent to single symbols under the embedding into
K
n
F
/
2
m
+
1
K
n
F
K_n F/2^{m+1} K_n F
is at most
2
n
−
1
2^{n-1}
under the assumption that
F
⊇
μ
2
m
+
1
F \supseteq \mu _{2^{m+1}}
. Since
K
2
F
/
2
m
K
2
F
≅
2
m
B
r
(
F
)
K_2 F/2^m K_2 F \cong {_{2^m}Br(F)}
for
n
=
2
n=2
, this coincides with the upper bound of
2
2
(proved by Tignol in 1983) for the symbol length of central simple algebras of exponent
2
m
2^m
that are Brauer equivalent to a single symbol algebra of degree
2
m
+
1
2^{m+1}
. The cases where the embedding into
K
n
F
/
2
m
+
1
K
n
F
K_n F/2^{m+1} K_n F
is of symbol length
2
2
,
3
3
, and
4
4
(the last when
n
=
2
n=2
) are also considered. The paper finishes with the study of the symbol length for classes in
K
3
/
3
m
K
3
F
K_3/3^m K_3 F
whose embedding into
K
3
F
/
3
m
+
1
K
3
F
K_3 F/3^{m+1} K_3 F
is one symbol when
F
⊇
μ
3
m
+
1
F \supseteq \mu _{3^{m+1}}
.