Here is my answer:
Concerning question 1, you can assume basic arithmetic rules and all the statements we mentioned in the lectures so far as true (for the statement in the book please provide precise reference :)). If I required you to prove the rules of calculations for exponents, it will take hours to solve this exercise: you would have to construct the natural numbers, you will have to recursely define addition, multiplication. and show that they satisfy the usual rules of calculations by using induction. Then you define power and prove the rule, again using induction... too much work :) especially since the focus of this course is not logic, or foundations of mathematics, but analysis.
Concerning question 1, you can assume basic arithmetic rules and all the statements we mentioned in the lectures so far as true (for the statement in the book please provide precise reference :)). If I required you to prove the rules of calculations for exponents, it will take hours to solve this exercise: you would have to construct the natural numbers, you will have to recursely define addition, multiplication. and show that they satisfy the usual rules of calculations by using induction. Then you define power and prove the rule, again using induction... too much work :) especially since the focus of this course is not logic, or foundations of mathematics, but analysis.
Concerning question 2: No! The use of mathematical notation will not influence your grade. Writing style differs a lot from subject to subject, but I come from a school in which quantifiers are good on notes and blackboards, but not when one typewrites. Thus I would much prefer one to use words instead of symbols when writing. The same is true for arrows. I prefer "If... Then" to $\Rightarrow$ and "if, and only if,..." to $\Leftrightarrow$. In the slides you migt see me using "iff", instead of "if, and only if...". This is because I was a bit lazy in typing them. You will never see me write iff in an article, grant application or similar :)
And if you ever ask me a thesis, or I am the gränskare of your thesis, I will complain a lot about the use of symbols arrows and iff in the text :) (so that you are prepared)
I hope this helps
Sofia