let $\pi$ be a cuspidal automorphic representation for gl(3) over a number field. we fix an integer $k \geq 2$ and we assume that the symmetric $m$th power lifts of $\pi$ are automorphic for $m \leq k$, cuspidal for $m < k$, and that certain associated rankin–selberg products are automorphic. in this setting, we bound the number of cuspidal isobaric summands in the $k$th symmetric power lift. in particular, we show it is bounded above by 3 for $k \geq 7$, and bounded above by 2 when $k \geq 19$ with $k$ congruent to 1 mod 3. we will also discuss the analogous problem for gl(4).