I was chatting with an older friend of mine when I happened to mention the ages of my two children. He said to me, "when your youngest was born your age was the square of the older one. Now your age is the product of both their ages."

I then thought about his age and the ages of his children and replied, "When your youngest was born your age was the product of the older two. Later on your age was the product of the oldest and youngest. Now your age is the product of the younger two."

What are our two ages?

(Assume we all have the same birthday.)