1. **State the problem:** Solve for $n$ in the equation $$\frac{x^5 \cdot x^n}{x^8} = x^{20}.$$\n\n2. **Recall the exponent rules:** When multiplying powers with the same base, add the exponents: $$x^a \cdot x^b = x^{a+b}.$$ When dividing powers with the same base, subtract the exponents: $$\frac{x^a}{x^b} = x^{a-b}.$$\n\n3. **Apply the multiplication rule in the numerator:** $$x^5 \cdot x^n = x^{5+n}.$$\n\n4. **Apply the division rule:** $$\frac{x^{5+n}}{x^8} = x^{(5+n)-8} = x^{n-3}.$$\n\n5. **Set the expression equal to the right side:** $$x^{n-3} = x^{20}.$$\n\n6. **Since the bases are the same and nonzero, equate the exponents:** $$n - 3 = 20.$$\n\n7. **Solve for $n$:** $$n = 20 + 3 = 23.$$\n\n**Final answer:** $$\boxed{23}.$$