1. **Problem statement:** Determine whether the expression $$\frac{\sqrt{38}}{a}$$ is rational or irrational, given that $$a$$ is a non-zero rational number. 2. **Recall definitions:** - A rational number can be expressed as $$\frac{p}{q}$$ where $$p$$ and $$q$$ are integers and $$q \neq 0$$. - An irrational number cannot be expressed as a ratio of two integers. 3. **Analyze the numerator:** $$\sqrt{38}$$ is irrational because 38 is not a perfect square. 4. **Analyze the denominator:** $$a$$ is rational and non-zero. 5. **Division of irrational by rational:** If $$r$$ is irrational and $$q$$ is rational and non-zero, then $$\frac{r}{q}$$ is irrational. 6. **Apply to our expression:** Since $$\sqrt{38}$$ is irrational and $$a$$ is rational and non-zero, $$\frac{\sqrt{38}}{a}$$ is irrational. **Final answer:** $$\frac{\sqrt{38}}{a}$$ is irrational.