On january 1st, jerome concocts a rumor and tells beth. on january 2nd, they each tell a new person. by the end of january 2nd, four people have heard the rumor. on each subsequent day, each person who has heard the rumor spreads it to a new person.
how many people have heard the rumor by the end of january 8th?

So we want to know how many people have heard the rumor if the rumor started on january 1st and has spread exponentially, 2^n, where n>0 and a whole number and the rumor was spreading till january 8th. So on the january 1st 2 people knew the rumor and the rumor was spreading for 8 days. That means we have 2^n people who know the rumor where 2 is the number of people each person tells the rumor and 8 is the number of days the rumor was spreading. so 2^8 is 256. So 256 people heard the rumor.