<s>[INST] Intersection multiplicity 
Let $f=y^2-x^3$ and $g=y^3-x^7$. Calculate the intersection multiplicity of $f$ and $g$ at $(0,0)$. 

I know the general technique for this (passing to the local ring) but I having difficulty with the fact that $3,7,2$ have no common factors.
 [/INST] I think you no longer need an answer, but I’d like to write something here since it may help some other students.
We need four lemmas here(You can find them in lecture 2 of Andreas Gathmann’s notes in Plane Algebraic Curves):
Lemma (1): For any three plane curves F, G, H, $$\mu_p(F, G)=\mu_p(F, G+FH)$$
Lemma (2): For any three plane curves F, G, H, $$\mu_p(F, GH)=\mu_p(F, G)+\mu_p(F, H)$$
Lemma (3): $\mu_p(F, G)\ge 1$ iff $P\in F\cap G$.
Lemma (4): $\mu_p(F, G)=1$ iff $<F, G>=I_p$, where $I_p =\{f/g: f, g\in K[x, y], f(P)=0, g(P)\neq 0\}$.
Now we can compute $\mu_0(y^2-x^3, y^3-x^7)$.
$\begin{align}\mu_0(y^2-x^3, y^3-x^7)\overset{by (1)}{=}\mu_0(y^2-x^3, y^3-x^7-y(y^2-x^3))=\mu_0(y^2-x^3, x^3y-x^7)\overset{by (2)}{=}\mu_0(y^2-x^3, x^3)+\mu_0(y^2-x^3, y-x^4)\overset{by (1)}{=}\mu_0(y^2-x^3+x^3, x^3)+\mu_0(y^2-x^3-y(y-x^4), y-x^4))=\mu_0(y^2, x^3)+\mu_0(x^4y-x^3, y-x^4)\overset{by (2), (2), respectively}{=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=}6\mu_0(y, x)+\mu_0(x^3, y-x^4)+\mu_0(xy-1, y-x^4)\overset{by (4), (1), (3), respectively}{=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=}6+\mu_0(x^3, y-x^4+x(x^3))+0=6+\mu_0(x^3, y)\overset{by (2)}{=}6+3\mu_0(x, y)\overset{by (4)}{=}6+3=9\end{align}.$
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