Nitaj [Ni95] has proved that there are infinitely many triples of coprime $3$-powerful numbers $a,b,c$ such that $a+b=c$, such as \[2^3\cdot 3^5\cdot 73^3+271^3 = 919^3.\] In Nitaj's construction at least two of $a,b,c$ are perfect cubes. Cohn [Co98] constructed infinitely many such triples, none of which are perfect cubes.

Euler had conjectured that the sum of $k-1$ many $k$th powers is never a $k$th power, but this is false for $k=5$, as Lander and Parkin [LaPa67] found \[27^5+84^5+110^5+133^5=144^5.\]

Cambie has found several examples of the sum of $r-2$ coprime $r$-powerful numbers being itself $r$-powerful. For example when $r=5$ \[3^761^5=2^83^{10}5^7+2^{12}23^6+11^513^5.\] Cambie has also found solutions when $r=7$ or $r=8$ (the latter even with the sum of $5$ $8$-powerful numbers being $8$-powerful).