Suppose that you have $1500 invested at 4% APR compounded
continuously. The balance of the account after t years is given by
A(t)=1500e^.o4t
Find the balance of the account after:
1year
3years and 3 months
I need to show all work!

What work?  You just plug 1 and 3.25 into your formula:

1500 * e^(0.04*1) = 1500 * e^(0.04) = 1500 * 1.040810774 = 1561.2161613

1500 * e^(.04*3.25) = 1500 * e^(.13) = 1500 * 1.138828383 = 1708.2425749

This is the answer I came up with but I wanted a second opinion to
verify that I was correct.

Might I ask where these questions are coming from?  They sound like
homework problems, and if you are older than middle school age (and do
not suffer from a developmental disability or other condition which
would cause you to deviate from "normal" development) you would be
better served giving your money to a tutor or a book publisher and
learning this yourself.  The various standard equations for a line and
evaluating simple functions should be within your grasp at that age;
if they are not, you will find it very difficult to progress in
mathematics through high school and/or college.

As someone who has taken an awful lot of math, I can authoritatively
say that a strong grasp of basic algebra is absolutely essential to
progress through geometry, trigonometry, calculus, and beyond. 
Algebra is the language in which these classes are taught and without
a firm understanding of it, you cannot effectively understand the
principles, even if you learn the formulas by rote and can solve
problems using them.

Whatever path you choose, I wish you the best of luck - mathematics is
a wonderful world to explore!

-Haversian