Suppose that $f(x)$ and $g(x)$ are polynomials of degree $4$ and $5$ respectively.  What is the degree of $f(x^3) \cdot g(x^2)$?
I need to find the degree of a product of two polynomials, which means I need to add their degrees. But before I do that, I need to adjust their degrees based on the powers of $x$ they are raised to. For $f(x^3)$, I need to multiply the degree of $f(x)$ by $3$, since every term of $f(x)$ will have its power of $x$ tripled. Similarly, for $g(x^2)$, I need to multiply the degree of $g(x)$ by $2$, since every term of $g(x)$ will have its power of $x$ doubled. So the degree of $f(x^3)$ is $4 \times 3 = 12$, and the degree of $g(x^2)$ is $5 \times 2 = 10$. Now I can add these degrees to get the degree of their product: $12 + 10 = 22$. 