When you do something shrewdly, you act in a particularly smart and savvy way. Negotiate shrewdly at a yard sale and you might get that cool vintage lunchbox for only five dollars!
Actions that require practical intelligence have to be done shrewdly. You might shrewdly decide to ask your brother for a favor after you bake him his favorite cookies, wisely guessing he'll be more likely to say yes. You could also turn down a job offer, shrewdly waiting until a better job is available. Shrewdly is from shrewd, which has been defined as "wise" for centuries but originally meant "wicked or depraved."