There are 2 ways of answering the 1st true-false question.
There are 2 ways of answering the 2nd true-false question.
...
There are 2 ways of answering the 8th true-false question.

That's 2x2x2x2x2x2x2x2 = 28 ways of answering the 8 
true-false questions.

For each of those ways of answering the true-false questions, 

There are 4 ways of answering the 1st multiple choice question.
There are 4 ways of answering the 2nd multiple choice question.
...
There are 4 ways of answering the 7th multiple choice question.

That's 3x3x3x3x3x3x3 = 37 ways of answering the 8 
true-false questions.

Answer: 2847 = 4194304 ways.

Edwin