If I am offered a full-time job that offers a yearly salary of 50,310 and 3 weeks paid vacation. I will not be paid for hours worked above 40 hours per week. So I estimate that I will work an average of 45 hours per week. What will be gross earnings per week? And what is the hourly wage for a 45-hour work week

There are 52 weeks in a year, plus 1 additional day (or 2 if it's a leap year), so your earnings per week will be approximately $50,310/52. Depending on how your employer runs payroll (monthly, semi-monthly, bi weekly) the actual amount will vary.

For the equivalent hourly wage, if you want to assume a 45 hour work week and 3 weeks paid vacation then you have $50,310/49/45 per hour. In this calculation the 3 weeks would include not only vacation but also holidays and any other "personal days" you may be allowed to take off.

Ok, I will assume this is homework not a real job, but if real, you may well expect to work over 45 hours, you need to clarify that. Many managers work 50 to 60 hours