To find the least common multiple (LCM) of two algebraic expressions, we need to consider the highest powers of all the variables involved in both expressions. Let's break down the given expressions:

1. First Expression: $4u^3w^2v^6$

   • Coefficient: 4
   • $u^3$
   • $w^2$
   • $v^6$

2. Second Expression: $22u^4w^8$

   • Coefficient: 22
   • $u^4$
   • $w^8$
   • No $v$ term

Steps to Find the LCM:

1. LCM of the Coefficients:

   • The coefficients are 4 and 22.
   • The LCM of 4 and 22 is 44.

2. LCM of the Variables:

   • For $u$: The highest power between $u^3$ and $u^4$ is $u^4$.
   • For $w$: The highest power between $w^2$ and $w^8$ is $w^8$.
   • For $v$: The highest power is $v^6$ (since the second expression does not have $v$).

Putting it all together:

The LCM is $44u^4w^8v^6$.

Would you like further details or have any questions? Here are 5 related questions to explore:

1. How do you find the LCM of numerical values only?
2. What is the significance of finding the LCM in algebraic expressions?
3. How does the LCM of algebraic expressions differ from the LCM of numbers?
4. Can you find the greatest common divisor (GCD) of these two expressions?
5. How would the LCM change if a new variable was introduced in one of the expressions?

Tip: When finding the LCM of algebraic expressions, always take the highest power of each variable from the expressions.