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All courses Algebra I · A-SSE.3.c 59 of 76
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.
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Hint

Write the month input in terms of years: m = 12y.

A yearly input must use one multiplier for a full year. Since 12 months make 1 year, combine 12 monthly factors into the yearly factor.

Solution walkthrough

01

Relate months and years

\[m=12y\]

One year contains 12 months, so y elapsed years correspond to 12y elapsed monthly intervals.

02

Substitute the converted input

\[\begin{aligned} P=100(1.02)^m \\ P(y)=100(1.02)^{12y} \end{aligned}\]

Replace the monthly interval count m by 12y. The monthly growth factor 1.02 remains the same inside the power.

03

Regroup the exponent

\[\begin{aligned} a^{\text{bc}}=(a^b)^c \\ (1.02)^{12y}=((1.02)^12)^y \end{aligned}\]

The power property groups the 12 monthly factor applications into one yearly factor, then applies that yearly factor y times.

04

State and interpret the yearly model

\[\begin{aligned} P(y)=100((1.02)^12)^y \\ \text{one}-\text{year}~\text{growth}~\text{factor}=(1.02)^12 \end{aligned}\]

The coefficient remains 100, while the new base is the compounded factor for 12 months. Substituting y=1 confirms one year uses 12 monthly multipliers.

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Another way

  1. Write twelve copies of 1.02 for each year and group those copies as one annual multiplier.

!

Common mistake

Do not multiply 1.02 by 12. Compounding over 12 months raises the monthly factor to the twelfth power.

Solution walkthrough video