Graduate tuition rates depend on whether you are a Washington State resident or not: non-residents pay tuition at a much higher rate. To qualify as a resident, assuming you are financially independent of your parents, you must have actually resided in the state for the preceding 12 months. Such things as voter registration, driver's license, vehicle registration, and bank accounts in the state are important for demonstrating residency.
For more information, see the following University of Washington web pages:
If you are offered financial aid in the form of a teaching assistantship, research assistantship, or fellowship, you will also be offered a waiver of tuition (except for approximately $450 per quarter in fees).
International applicants will be asked to show evidence that they have sufficient resources to cover their cost of living plus tuition for the first year, which in recent years has amounted to more than US $40,000. If the applicant is offered support by the Mathematics Department, the usual Teaching Assistantship salary and tuition waiver will cover most of this amount. The applicant might need to provide a small additional amount from other sources, however.