Resolution of the (2,5) cusp by two blowups ($(x,y)=(t^5,t^2)⟼(u,y)=(t^3,t^2)⟼(v,y)=(t,t^2)$)

Blow up the origin and use the chart $x=uy$, giving the strict transform $u^2-y^3=0$. Blow up its remaining singular point and use $u=vy$, giving $v^2-y=0$, which is smooth because its derivative in $y$ is $-1$. The complementary chart at each stage has no point of the strict transform on its exceptional divisor. These two blowups of a smooth algebraic surface resolve the branch, although the smooth transform is still tangent to an exceptional divisor.