Find the greatest common factor of these three expressions. $27 u^{5}, 63 u^{4}$, and $18 u$

Step 1 ：Given the expressions are $27 u^{5}, 63 u^{4}$, and $18 u$.

Step 2 ：We need to find the greatest common factor (GCF) of these expressions. The GCF of a set of numbers is the largest number that divides evenly into each of the numbers. It can be found by listing the prime factors of each number, and then multiplying the common factors.

Step 3 ：For the given expressions, we need to find the GCF of the coefficients and the powers of $u$ separately.

Step 4 ：The coefficients are 27, 63, and 18. The powers of $u$ are 5, 4, and 1.

Step 5 ：The greatest common factor of the coefficients 27, 63, and 18 is 9. The greatest common factor of the powers of $u$ (5, 4, and 1) is 1.

Step 6 ：Therefore, the greatest common factor of the expressions $27 u^{5}, 63 u^{4}$, and $18 u$ is $9 u^{1}$.

Step 7 ：Final Answer: The greatest common factor of the expressions $27 u^{5}, 63 u^{4}$, and $18 u$ is \(\boxed{9 u}\).