Master form ($A_0 b^{t/p}$)
SAT glossary · Course Foundations
Definition
The single exponential model that subsumes the whole table: start at A_0 and multiply by b every p units of t. Half-life is b=1/2, doubling is b=2, annual growth is b=1+r with p=1. The period always goes in the DENOMINATOR of the exponent — the distractor M_0(0.88)^3t inverts it, because multiplying is the reflex and dividing is correct.
Master form ($A_0 b^{t/p}$) in plain English
This is the one master template that all the different exponential-growth-and-decay word problems on the test collapse into: A_0 b^t/p means 'start at the value A_0, and multiply by b every time p units of time pass.' Different real-world scenarios just plug in different values: half-life problems use b=1/2, something doubling uses b=2, and steady annual growth at rate r uses b=1+r with p=1 (since it happens every single year). The one thing to burn into memory: the time PERIOD p always belongs in the DENOMINATOR of the exponent, dividing t — a wrong answer choice often flips this and puts the period in as a multiplier instead (like M_0(0.88)^3t instead of the correct M_0(0.88)^t/3), because multiplying just feels more natural than dividing, even when it's wrong.
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