GOZUNTA
Try it now
All courses Math II · A-SSE.3.c 13 of 73
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 so the input is measured in years instead of months.
Your answer
With a free account

Find a specific skill faster.

Search the curriculum instead of wandering through random problems until the right kind appears.

Create a free account
With paid access

Keep going when the starter set ends.

A subscription removes the locked gaps from your selected course so practice can continue through the full sequence.

Compare plans

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 to years

\[m~=~12y\]

The original exponent m counts months. If y counts years, then y years contains 12y months.

02

Substitute the new time expression

\[P(y)~=~100(1.02)^(12y)\]

Substitute 12y for the month input so the expression uses the year variable.

03

Rewrite the exponent structure

\[100(1.02)^(12y)~=~100((1.02)^12)^y\]

Use abc=(ab)c to group the 12 monthly multipliers into one yearly multiplier.

04

State the model in years

\[P(y)=100((1.02)^12)^y\]

Now y counts years, and the base (1.02)¹2 represents the growth factor for one year.

+

Another way

  1. Check one year directly: after 1 year, the original monthly model uses m=12, so P=100(1.02)¹2. The rewritten model should give the same result when y=1.

  2. You can also view the new base as the yearly multiplier. Twelve monthly multipliers of 1.02 combine to make (1.02)¹2.

!

Common mistake

A common mistake is to change the input letter to y but leave the base as 1.02. That would make the model grow by only 2% each year instead of applying twelve 2% monthly growth factors during one year.