A very common problem when it comes to camera's is that the lens gets stuck and doesn't turn on as a result of that. The cause is very likely dust that is stuck in the lens cap.

Don' t despair, you might be able to fix it yourself (assuming that it's not under warrenty anymore and it doesn't pay to have it professionally fixed).

Check out this link- I used it and found it very helpful. (I successfully fixed a camera :).