Answer
$2,037,468,266,496$ different passwords.
Work Step by Step
First character (a letter): 26 choices
Second character (a letter or a digit): $26+10=36$ choices
Third character (a letter or a digit): $26+10=36$ choices
Fourth character (a letter or a digit): $26+10=36$ choices
Fifth character (a letter or a digit): $26+10=36$ choices
Sixth character (a letter or a digit): $26+10=36$ choices
Seventh character (a letter or a digit): $26+10=36$ choices
Eighth character (a letter or a digit): $26+10=36$ choices
Using the Multiplication Rule of Counting (see page 298):
$26\times36\times36\times36\times36\times36\times36\times36=26\times36^7=2,037,468,266,496$
