Rewrite the exponential growth model expression $y = 450 \cdot (1.08)^{t/3}$, where $t$ is measured in years, into the standard equivalent annual compounding form $y = a \cdot (1 + r)^t$, rounding the effective annual growth rate $r$ to four decimal places.

Rewrite the exponential growth model expression $y = 450 \cdot (1.08)^{t/3}$ , where $t$ is measured in years, into the standard equivalent annual compounding form $y = a \cdot (1 + r)^t$ , rounding the effective annual growth rate $r$ to four decimal places.

Final answer

$y = 450 \cdot (1 + 0.0260)^t$

Steps

Separate the exponent $t$

To rewrite the given exponential expression into the form $y = a \cdot (1 + r)^t$ , we isolate the exponent $t$ using the power of a power property of exponents: $(b^m)^n = b^{m \cdot n}$ .

Rewrite the exponent $\frac{t}{3}$ as $\frac{1}{3} \cdot t$ : $y = 450 \cdot (1.08)^{\frac{t}{3}} = 450 \cdot \left((1.08)^{1/3}\right)^t$

Calculate the new annual growth factor

Evaluate the base $(1.08)^{1/3}$ : $(1.08)^{1/3} = \sqrt[3]{1.08} \approx 1.02598557\dots$

Determine the annual growth rate $r$

The annual growth factor relates to the annual growth rate $r$ by: $1 + r = (1.08)^{1/3}$ $r = (1.08)^{1/3} - 1 \approx 0.02598557\dots$

Rounding $r$ to four decimal places gives: $r \approx 0.0260$

Write in standard form

Substitute the initial value $a = 450$ and the rate $r = 0.0260$ back into the formula: $y = 450 \cdot (1 + 0.0260)^t$

Common Pitfall: A frequent mistake is dividing the growth rate $0.08$ directly by $3$ (giving $\approx 0.0267$ ). However, because compounding is multiplicative rather than additive, the correct rate requires taking the cube root, not dividing by $3$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student multiplies the base or exponent factor by 1/3 instead of taking the cube root of the growth factor.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
1 + r = 1 + $\frac{0.08}{3}$ $\implies$ 1 + r $\approx$ 1.0267
Correct
1 + r = (1.08)^{1/3} $\approx$ 1.0260
Because exponential growth compounds multiplicatively, converting an exponent of $\frac{t}{3}$ into $t$ requires taking the cube root of the base: $(1.08)^{t/3} = ((1.08)^{1/3})^t$ . Dividing the nominal rate by $3$ assumes linear, simple interest rather than compound growth.
Step 2
r $\approx$ 0.0267
Correct
r $\approx$ 0.0260
You correctly subtracted $1$ from your computed base, but the resulting rate inherited the error from dividing by $3$ in the previous step.
Step 3
y = 450(1 + 0.0267)^t
Correct
y = 450 $\cdot$ (1 + 0.0260)^t
You assembled the standard form $y = a(1 + r)^t$ correctly using your calculated rate, but the final value reflects the earlier calculation error.

WHAT YOU DID WELL

Compliments from Your Teacher

You identified the goal of rewriting the expression into the standard annual compounding form $y = a(1 + r)^t$ .

You preserved the initial value $a = 450$ accurately in the final equation.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Linear rate division instead of radical exponentiation

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated compounding growth as an additive or linear process by dividing the rate by 3 ( $r = \frac{0.08}{3}$ ), rather than applying the power-of-a-power property $b^{t/k} = (b^{1/k})^t$ to find the annual multiplier via roots.

REASONING PATH

Your ACEpath teacher’s suggested approach

Power of a Power Rule: For any base $b$ and fractional exponent, $b^{t/n} = \left(b^{1/n}\right)^t = (\sqrt[n]{b})^t$ .
Always isolate the standalone exponent $t$ on the outside by taking the $n$ -th root of the original base factor.
Base vs. Rate Conversion: In exponential models, the entire base $(1 + R)$ scales exponentially over time, so $1 + r = (1 + R)^{1/n}$ , not $1 + \frac{R}{n}$ .
Find the annual growth factor $1 + r$ by evaluating $(1 + R)^{1/n}$ before subtracting $1$ to find the rate $r$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Apply the power of a power property to isolate $t$ : $y = 450 \cdot \left((1.08)^{1/3}\right)^t$ .
Compute the effective 1-year growth factor: $(1.08)^{1/3} \approx 1.025986$ .
Solve for the annual rate $r$ : $r = 1.025986 - 1 \approx 0.0260$ .
Substitute into the annual compounding form: $y = 450 \cdot (1 + 0.0260)^t$ .

Final answer

y = 450 $\cdot$ (1 + 0.0260)^t

NEXT STEPS

Final Takeaways from Your Teacher

Practice converting exponential expressions with fractional exponents into standard forms by applying the power property $(b^{m/n}) = (b^m)^{1/n}$ .
Check that compound interest factors change multiplicatively via roots or powers, rather than dividing or multiplying the rate linearly.

OverviewACEpath Teacher's Diagnosis

02 The student misinterprets the exponent t/3 as raising the base to the power of 3 instead of taking the 1/3 power.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
y = 450 $\cdot$ ((1.08)^3)^t
Correct
y = 450 $\cdot$ ((1.08)^{1/3})^t
The exponent is $\frac{t}{3}$ , which factors as $\frac{1}{3} \cdot t$ . Using the power of a power property $(b^m)^n = b^{m \cdot n}$ , the inner base must be raised to the power of $\frac{1}{3}$ (the cube root), not $3$ .
Step 2
(1.08)^3 $\approx$ 1.2597
Correct
(1.08)^{1/3} $\approx$ 1.0260
You calculated $(1.08)^3$ accurately, but the correct operation requires evaluating $(1.08)^{1/3} = \sqrt[3]{1.08}$ .
Step 3
1 + r = 1.2597 $\implies$ r $\approx$ 0.2597
Correct
1 + r $\approx$ 1.0260 $\implies$ r $\approx$ 0.0260
You correctly solved for $r$ by subtracting $1$ from your growth factor, but the numerical value was carried over from the earlier exponent mistake.
Step 4
y = 450(1 + 0.2597)^t
Correct
y = 450(1 + 0.0260)^t
You correctly assembled the final expression format $y = a(1 + r)^t$ , but the rate $r$ reflects the initial exponent error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly identified that the exponent needs to be factored so that $t$ sits outside the parentheses.

You correctly related the annual growth factor $1 + r$ to the rewritten base.

You accurately subtracted $1$ to isolate the rate $r$ and maintained the initial coefficient $450$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Fractional exponent inversion

You inverted the division by $3$ in the exponent into a multiplication by $3$ , writing $(1.08)^3$ instead of $(1.08)^{1/3}$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing division in an exponent with multiplication: $\frac{t}{k}$ means multiplying $t$ by the unit fraction $\frac{1}{k}$ , which corresponds to taking a root ( $b^{1/k}$ ), not raising to the integer power ( $b^k$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Fractional Exponent Decomposition: For any expression $b^{\frac{t}{k}}$ , separate the variable factor: $\frac{t}{k} = \frac{1}{k} \cdot t$ , so $b^{\frac{t}{k}} = \left(b^{1/k}\right)^t$ .
Always rewrite division by $k$ in the exponent as multiplication by $\frac{1}{k}$ before applying power-of-a-power.
Reasonableness Check on Growth Rates: If an item grows by $8\%$ every $3$ years, the annual growth rate must be smaller than $8\%$ , not larger.
Verify whether the time conversion makes the interval shorter or longer: a shorter compounding period should yield a smaller periodic rate.

CORRECTED PATH

ACEpath Teacher's Demonstration

Rewrite $\frac{t}{3}$ as $\frac{1}{3} \cdot t$ : $y = 450 \cdot \left((1.08)^{1/3}\right)^t$ .
Calculate the annual growth factor: $(1.08)^{1/3} = \sqrt[3]{1.08} \approx 1.025986$ .
Find the annual rate $r$ : $1 + r \approx 1.025986 \implies r \approx 0.0260$ .
Substitute into standard form: $y = 450 \cdot (1 + 0.0260)^t$ .

Final answer

$y = 450 \cdot (1 + 0.0260)^t$

NEXT STEPS

Final Takeaways from Your Teacher

Practice separating rational exponents into unit fractions times the variable, such as $\frac{t}{4} = \frac{1}{4} \cdot t$ .
Remember to evaluate fractional exponents using roots on your calculator: $b^{1/n} = \sqrt[n]{b}$ .
Perform a sanity check on growth rates to ensure annual rates align logically with multi-year or fractional-year rates.

OverviewACEpath Teacher's Diagnosis

03 The student rounds the growth rate to four decimal places incorrectly by shifting the decimal point as if converting to a percentage without adjusting the rounding.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$y = 450 \cdot ((1.08)^{\frac{1}{3}})^t \implies 1.08^{\frac{1}{3}} \approx 1.0259855$
Step 2
$1 + r = 1.0259855 \implies r = 0.0259855$
Step 3
$r \approx 0.0259$
Correct
$r \approx 0.0260$
When rounding $0.0259855\dots$ to four decimal places, the fifth digit is $8$ , which is $\ge 5$ . This requires rounding up the fourth digit ( $9$ ), carrying over to give $0.0260$ rather than truncating to $0.0259$ .
Step 4
$y = 450(1 + 0.0259)^t$
Correct
$y = 450(1 + 0.0260)^t$
This step correctly substitutes your computed rate into the formula, but carries forward the rounding error from Step 3.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly used the power of a power exponent rule to isolate the exponent $t$ .

You calculated the cube root of the growth factor accurately.

You correctly related the annual growth factor $1+r$ to the growth rate $r$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Rounding / Truncation Error

You truncated the decimal at the fourth decimal place instead of rounding up based on the fifth digit ( $8$ ).

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking carry-over rounding when the target decimal digit is $9$ and the subsequent digit is $\ge 5$ , leading to accidental truncation.

REASONING PATH

Your ACEpath teacher’s suggested approach

Identify the deciding digit: To round to $n$ decimal places, inspect the digit in the $(n+1)$ -th place.
For four decimal places, check the fifth digit: in $0.02598\dots$ , the fifth digit is $8$ .
Carry over when rounding a nine: If the deciding digit is $5$ or greater, add $1$ to the preceding digit; $9$ rounds up to $10$ , which carries $1$ into the next place to the left.
Add $1$ to the fourth place: $0.0259 + 0.0001 = 0.0260$ , preserving the trailing zero to show four decimal places.

CORRECTED PATH

ACEpath Teacher's Demonstration

Inspect the fifth decimal digit of $r = 0.0259855\dots$ : the digit is $8$ .
Because $8 \ge 5$ , round up the fourth digit ( $9$ ), which carries into the third digit: $r \approx 0.0260$ .
Substitute $r = 0.0260$ and $a = 450$ back into $y = a(1 + r)^t$ .

Final answer

$y = 450 \cdot (1 + 0.0260)^t$

NEXT STEPS

Final Takeaways from Your Teacher

Double-check whether you are rounding or truncating by circling the deciding digit right after your target decimal place.
Remember to retain trailing zeros when rounding to a specified number of decimal places (e.g., $0.0260$ , not $0.026$ ).
Practice rounding decimals with trailing nines that trigger carry-over.

Lock it in

Question

How do you rewrite $b^{t/k}$ into the equivalent form $(B)^t$ ?

Answer

$B = b^{1/k} = \sqrt[k]{b}$ . Compounding is multiplicative, so fractional exponents become roots of the base.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student multiplies the base or exponent factor by 1/3 instead of taking the cube root of the growth factor.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student misinterprets the exponent t/3 as raising the base to the power of 3 instead of taking the 1/3 power.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student rounds the growth rate to four decimal places incorrectly by shifting the decimal point as if converting to a percentage without adjusting the rounding.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

Similar question