Is two thirds and four ninths equivalent?

Umm, no. Two thirds and six ninths are equivalent. They look like this 2/3 and 6/9, to get to 9 you times the 3 by 3 so to get the 6 you have to times the 2 by 3. I can see that you've seen that you have squared 3 so you had to square 2, but unfortunately that isn't what happens.