Find the least common multiple of these two expressions. \[ 4 u^{8} y^{4} \text { and } 22 w^{5} u^{2} y^{7} \]

Step 1 ：Given the expressions \(4u^{8}y^{4}\) and \(22w^{5}u^{2}y^{7}\).

Step 2 ：The least common multiple (LCM) of two expressions is the smallest expression that is a multiple of both expressions. To find the LCM of two expressions, we need to take the highest power of each variable from both expressions.

Step 3 ：For the variable \(u\), the highest power is \(8\) from the first expression.

Step 4 ：For the variable \(y\), the highest power is \(7\) from the second expression.

Step 5 ：The second expression also has a variable \(w\) with power \(5\), which is not present in the first expression.

Step 6 ：The coefficients of the expressions are \(4\) and \(22\). The LCM of these two numbers can be found by multiplying them together if they are coprime, which they are in this case.

Step 7 ：So, the LCM of these two expressions is \(4*22*u^{8}*w^{5}*y^{7}\).

Step 8 ：Final Answer: The least common multiple of the two expressions \(4u^{8}y^{4}\) and \(22w^{5}u^{2}y^{7}\) is \(\boxed{44u^{8}w^{5}y^{7}}\).