I had a question about example 1 on page 245 in the PWN Math Guide fourth edition book. I thought the side AM is equal to 3√3 because of the 30-60-90 rule but according to the answer, AM is 4. I understand how it is 4 but how is it not 3√3?

Careful! It may look like a 30°-60°-90° triangle, but 1) you’re not given any angle measures to confirm that, and 2) the figure isn’t drawn to scale. The only route on this question is to use the Pythagorean theorem (and special 3-4-5 triples) to find the side lengths you need.