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Rewrite Exponential
Expressions

Learn how to use the properties of exponents to transform exponential functions and reveal hidden rates like monthly interest from annual growth.

Algebra 2

What Is an Exponential Expression?

An exponential expression has the general form \( a \cdot b^t \), where \( a \) is the initial value, \( b \) is the base (growth or decay factor), and \( t \) is the exponent representing time or another variable.

\( a \) : Initial Value
The starting amount when \( t = 0 \)
\( b \) : Base
Growth factor if \( b > 1 \), decay if \( 0 < b < 1 \)
\( t \) : Exponent
Often represents time (years, months, hours)

Why Rewrite Exponential Expressions?

The same exponential function can be expressed in different but equivalent forms. Rewriting lets you reveal information that is hidden in the original expression.

Change Time Units
Convert annual rates to monthly, daily, or hourly equivalents
Reveal Hidden Rates
Find the per-period percent change embedded in the expression
Compare Models
Put two expressions in the same units to compare growth speeds

Visualizing Exponential Growth

Time (years) Value 0 1 2 3 4 5 0 0.5 1.0 1.5 2.0 2.5 \( 1.15^t \) \( (1.15^{1/12})^{12t} \)

The Power of a Power Rule

The key property that makes all of this work:

This identity is a direct consequence of the exponent rule \( (x^a)^b = x^{ab} \). By choosing \( n \) to match a desired time unit, we can convert any exponential expression to a new period.

\( n = 12 \)
Years → Months
\( n = 4 \)
Years → Quarters
\( n = 365 \)
Years → Days

The Rewrite Template

To rewrite \( a \cdot b^t \) in terms of \( n \) sub-periods per original period:

1
Identify \( b \) and \( n \): \( b \) is the current base; \( n \) is how many sub-periods fit in one original period.
2
Compute \( b^{1/n} \): Take the \( n \)-th root of \( b \). This is the new base (sub-period growth factor).
3
Replace the exponent: Change \( t \) to \( nt \) so the units match. The expression becomes \( a \cdot (\text{new base})^{nt} \).

Rewriting vs. Re-Compounding

These two ideas look similar but are fundamentally different:

Rewriting (This Lesson) Re-Compounding
What changes? Only the form of the expression The value of the function changes
Are they equal? Yes: exact same function No: different function entirely
Example \( 1.15^t = (1.15^{1/12})^{12t} \) \( (1 + 0.15/12)^{12t} \)
Rate relationship Equivalent rate derived from original Nominal rate divided equally

How to Apply the Rewrite

1
Read the problem: Identify the current exponential expression \( a \cdot b^t \) and the target time unit (monthly, quarterly, daily, etc.).
2
Choose \( n \): Determine how many target units fit in one current period. For example, 12 months in 1 year means \( n = 12 \).
3
Compute \( b^{1/n} \): Use a calculator to find \( b^{1/n} \). Round to an appropriate number of decimal places.
4
Write the new form: \( a \cdot (b^{1/n})^{nt} \). The new base reveals the per-sub-period growth factor.
5
Interpret the rate: The equivalent per-period rate is \( b^{1/n} - 1 \). Convert to percent by multiplying by 100.

Example 1: Annual → Monthly

Rewrite \( 1.15^t \) to reveal the approximate monthly growth factor.

1
We have \( b = 1.15 \) and want months, so \( n = 12 \).
2
Compute: \( 1.15^{1/12} \approx 1.01171 \).
3
Rewrite: \( 1.15^t = (1.15^{1/12})^{12t} \approx 1.0117^{12t} \).

Example 2: Annual → Quarterly

A population grows by \( P(t) = 500 \cdot 1.08^t \), where \( t \) is in years. Find the quarterly growth factor.

1
Identify: \( a = 500 \), \( b = 1.08 \), target: quarters, so \( n = 4 \).
2
Compute: \( 1.08^{1/4} \approx 1.0194 \).
3
Rewrite: \( P(t) = 500 \cdot 1.0194^{4t} \).

Example 3: Decay → Daily Rate

A radioactive substance decays by \( M(t) = 200 \cdot 0.94^t \) (years). Find the daily decay factor.

1
Here \( b = 0.94 \) (decay, since \( b < 1 \)) and \( n = 365 \).
2
Compute: \( 0.94^{1/365} \approx 0.999831 \).
3
Rewrite: \( M(t) = 200 \cdot 0.999831^{365t} \).

Going the Other Direction

You can also rewrite from smaller to larger time units. If you know the monthly factor, you can find the annual factor.

If something grows by a factor of \( 1.005 \) each month, the annual factor is:

The Conversion Flowchart

Annual \( b^t \) Monthly \( (b^{1/12})^{12t} \) Quarterly \( (b^{1/4})^{4t} \) Daily \( (b^{1/365})^{365t} \) \( n=12 \) \( n=4 \) \( n=365 \) All expressions are exactly equal!

Common Mistakes

✕ Dividing the Rate
Writing \( (1 + r/n)^{nt} \) instead of \( (b^{1/n})^{nt} \). Dividing the rate by \( n \) gives a different function, not an equivalent rewrite.
✕ Forgetting to Adjust the Exponent
Writing \( (b^{1/n})^{t} \) instead of \( (b^{1/n})^{nt} \). You must multiply the exponent by \( n \) to compensate for the root.
✕ Wrong Direction on Rate
Saying the monthly rate is \( 15\% / 12 = 1.25\% \). The actual equivalent monthly rate is \( 1.15^{1/12} - 1 \approx 1.17\% \), which is slightly different.

Summary of Key Conversions

Original Target Unit Periods (\(n\)) New Form
\( b^t \) (annual) Monthly 12 \( (b^{1/12})^{12t} \)
\( b^t \) (annual) Quarterly 4 \( (b^{1/4})^{4t} \)
\( b^t \) (annual) Weekly 52 \( (b^{1/52})^{52t} \)
\( b^t \) (annual) Daily 365 \( (b^{1/365})^{365t} \)
\( b^m \) (monthly) Annual 12 \( (b^{12})^{t} \)

Question 1 of 5

Which expression is equivalent to \( 1.15^t \) and reveals the approximate monthly growth factor?

Question 2 of 5

A function is given by \( f(t) = 300 \cdot 1.06^t \), where \( t \) is in years. What is the approximate quarterly growth rate?

Question 3 of 5

A substance decays according to \( A(t) = 80 \cdot 0.90^t \) (years). Which rewrite reveals the daily decay factor?

Question 4 of 5

If a stock grows by a factor of \( 1.003 \) each week, what is its approximate annual growth rate?

Question 5 of 5

Why does rewriting \( b^t \) as \( (b^{1/n})^{nt} \) produce an equivalent expression?