$A = A_0(1 + r)^t$ — read aloud, A equals A-nought, times one plus r, to the t. It is one fact wearing three outfits: solve it for the final amount, for the starting amount, or for the rate, and it answers three different exam questions.

The exponent is where compounding lives. Growing 10% a year for 3 years is NOT 30% — it is $1.1^3 = 1.331$, a 33.1% gain, because year two grows on year one's winnings. The gap looks small at three years and enormous at thirty; that gap is the entire business model of a bank.

Two guard rails. The rate must arrive as a DECIMAL before it enters the bracket — a 12 where 0.12 belongs multiplies your answer by an absurd factor, and the size of the answer will tell you so instantly. And when you solve for $r$, the escape from the exponent is a ROOT: $r = \left(\dfrac{A}{A_0}\right)^{1/t} - 1$. The total growth divided by t is a tempting shortcut and a wrong one — it always overshoots, because it pretends the early years grew on the same base as the late ones.