Use rules of inference to show that the hypotheses "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on," "If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion "It rained."

1- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis.2- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in the first question?

Factor the polynomial, if possible. if the polynomial cannot be factored, write prime. 9n^3 + 27n^2 – 25n – 75

Emmanuel added 888 links per minute to his chain mail. allesia started 202020 minutes after emmanuel and added 131313 links per minute to her chain mail. how long had emmanuel worked when allesia caught up to him, and how many links had he added?