Common multiple question? How would I solve the following question
find the least common multiple of these two expression.
$14w^7y^2$ and $6w^4y^5x^8$ would I just have to multiply them. 
 A: $$14w^7y^2 = 2 \cdot 7 w^4w^3y^2 = (2w^4y^2)(7w^3)\tag{1}$$
$$6w^4y^5x^8 = 2 \cdot 3 w^4 y^2 y^3x^8 = (2 w^4y^2)(y^3x^8)\tag{2}$$
Least common multiple: $$(2w^4y^2)(7w^3)(3y^3x^8) \;= \;42w^7y^5x^8\tag{3}$$
Can you see how we could exclude the commong factor $2w^4y^2$, using it only once in the product? Each of $(1), (2)$ divides $(3)$. 
Can you make connections between the original appearance of $(1), (2)$ and the result, the least common multiple as shown on the right hand side of $(3)$?
A: Hint only: Consider for example the expressions: $xy^2$ and $xy$. In this case the least common multiple is $xy^2$ (and not $x^2y^3$) because this expression is the "smallest" that is both divisible by both $xy^2$ and $xy$. The least common multiple is the "smallest" expression that you can get from multiplying each of the two by something else. So
$$
xy^2 = 1\cdot (xy^2)\\
xy^2 = y\cdot(xy^2).
$$
The point is that you might have something that is "smaller" than the product. Another example is with numbers. The least common multiple of $10$ and $25$ is $2\cdot 5^2 = 50$ again because $50$ is the smallest number that is divisible by both $25$ and $50$. 
Now for your two expressions: $a = 14w^7y^2$ and $b = 6w^4y^5x^8$. Ask your self what the "smallest" expression that you can get from multiplying $a$ by something and $b$ by something (else). What is the "smallest" expression that is divisible by both $a$ and $b$? You see that in the final expression you will need to have a $x^8$ because of multiplying $b$ by anything is always going to have an $x^8$ in it. But $a$ and $b$ already both have some $w$'s in them. So maybe it is enough if you multiply $b$ by $w^3$ to get up to $w^7$.
A: Hint $\rm\ \ lcm(a^j b, a^k c) = a^k lcm(b,c)\ \ $ if $\rm\, \ j\le k,\ \ (a,b) = 1 = (a,c).\:$ So, e.g., for $\rm\:x,y,w\:$ pair-coprime
$$\rm lcm(w^7 y^2, w^4 y^5 x^8) = w^7 lcm(y^2,\,y^5 x^8) = w^7 y^5 lcm(1,x^8) = w^7 y^5 x^8 $$
A: $$14w^7y^2=2^1\times7^1w^7y^2$$
$$6w^4y^5x^8=2^1\times3^1w^4y^5x^8$$
Now just collect all the greatest powers for each multiplicand (not positive this is the proper term) yielding an answer of
$$2^1\times3^1\times7^1w^7y^5x^8=42w^7y^5x^8$$
