This equation's solutions can be classified into the following categories, or solved according to the Theorem:

If a partial word *w* exists such that *x, y, *and *z* are contained in powers of *w*, then the solution is **Trivial**, or **Type 1**.

If the partial words *x, y, *and *z* satisfy *x* ↑ *z* and *y* ↑ *z*, then the solution is a **Type 2** solution. Additionally, if *z* is a full word, then it is a **Trivial** solution.

Theorem: Let *x, y, *and *z* be *primitive partial words* such that (*x, z*) and (*y, z*) are good pairs. Let *m, n, *and *p* be integers such that *m* ≥ 2, *n* ≥ 2, and *p* ≥ 4. Then the equation *x*^{m}*y*^{n} ↑ *z*^{p} has only solutions of **Type 1** or **Type 2**, *unless* *x*^{2} ↑ *z*^{k}z_{p} for some integer *k* ≥ 2 and non-empty prefix *z*_{p} of *z*, *or* *z*^{2} ↑ *x*^{l}x_{p} for some integer *l* ≥ 2 and non-empty prefix *x*_{p} of *x*.