In the context of the word exercise, e.g. in a textbook, is it more common to say answer or solution?
It would depend what the exercise was; if it is a question such as "What is the capital of Mongolia?", the response is an answer. If the question were something like "find the length of the hypotenuse of this triangle", you could use either word; the workings would be part of the solution but not part of the answer.
I'd also suggest that an incorrect response to either question could be called an answer, but wouldn't be a solution.
Answer is the word you are looking for.
My dictionary says:
Answer • a thing written or said in reaction to a question in a test or quiz : write your answers on a postcard.
Like the definition says, it is what you say in response to the exercise's question, being it an actual question or a fill-in-the-gap type, such as:
Mary _____ (go) to the Mall yesterday.
Your answer would be "went" in this case.
What you call solution reminds me of the
solutions section at the end of the Textbook, but it is rather called Key than "solution".
Plus solution meant as "answer" is too general, since solution can be the "answer" to a problem and not just to an exercise.
Your choice of 'answer' or 'solution' would lie in what the exercise was.
The verb for exercise would most likely be 'complete' or simply do.
'Answer' would be used with a question and 'solution' with a problem.
Answer the question. -or- Provide an answer to the question. Solve the problem. -or- provide a solution to the problem.