For compound interest: Total amount after time t = A = P*(1+r/100)^t.

A bank account with $100 earns 8% annual interest, compounded quarterly. If there are no deposits or withdrawals, how much money will be in the account after 6 months?

There are 4 quarters each year. Each has 3 months in it.
Since interest is compounded quarterly, we need to find the quarterly rate.
Quarterly rate r= Annual Rate/(Number of Quarters in one year) = 8/4 = 2%.
We need to find the amount after 6 months, i.e. 2 quarters, so time t = 2.
Principal P = 100.

Now its easy.
Amount A = P*(1+r/100)^t = 100* (1+2/100)^2 = 100* 1.02^2 = 100* 1.0404 = 104.04