Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Rewrite an exponential model when the input unit changes
Problem
Rewrite \(P(y)=100(1.12)^y\) using \(m~\text{months}\). Set \(y=m/12\), express the result with a monthly base raised to \(m\), and explain why that base is the twelfth root of the yearly factor while \(100\) stays unchanged.
Big Picture
What this problem is really about
A month is one twelfth of a year, so changing the input unit requires replacing the yearly interval count by a fractional year count. We’ll substitute that conversion, regroup the exponent into a monthly base raised to the month count, and verify that twelve copies of the new factor recover the original annual multiplier while the zero-time value stays fixed.
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