Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Rewrite an exponential model when the input is measured in a different unit
Problem
Rewrite \(P(m)=100(1.02)^m\) using \(y~\text{years}\) as the input. Begin with \(m=12y\), substitute into the exponent, apply \(a^{\text{bc}}=(a^b)^c\), and identify the one-year growth factor.
Big Picture
What this problem is really about
Changing the input unit changes how many original growth intervals one new interval contains. We’ll express months as a multiple of years, substitute that count into the exponent, regroup the power into one compounded yearly factor raised to the year count, and keep the initial coefficient unchanged.
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