The president question was a beauty, so glad I spent 5 minutes deciding what one to do, there was so much to talk about.
Power of pursuasion is pork barrel spending and all that, then there's head of state singing treaties, commander in chief of armed forces, vetoes, executive orders, appointing judges, foreign policy.
There was sooo much talk about in this question and so many recent examples I crammed that I crammed I in
I said the power that is the most important depends on the political climate, make up of congress and something else, but power of persuasion is the most important as it is informal and doesn't have a check from other braches like the other powers do, also the power of persuasion allows the president to influence congress on domestic policy and legislation, something not enumerated and not intended by the constitution.