Prove that the language { uw : |u| =2 |w| } is regular
Any string of the form $uw$ where $|u| = 2|w|$ is a string whose length is a multiple of $3$. We are therefore considering the language of strings over some alphabet $\Sigma = \{ a_1, \ldots, a_k \}$ (not specified in the question). whose length is a multiple of $3$. Let $\Sigma$ be the regular expression $a_1 + \cdots + a_k$. Then a regular expression descring the language in question is

$$(\Sigma\Sigma\Sigma)^*$$