Transforming Exponential Expressions

The same exponential expression can be written with different bases, and each version reveals a different rate. $81^t$ and $3^{4t}$ are equal for every value of $t$ — but the second version lets you talk about a process that multiplies by $3$ , four times per unit of $t$ .

The tool for switching bases is the power of a power rule: $(b^c)^t = b^{ct}$ . Write the old base as a power of the new base, then multiply the exponents. The value of the expression never changes — only the way you read it.

The power of a power rule

The rule $b^{ct} = (b^c)^t$ works in both directions. Going one way, $\left(3^4\right)^t = 3^{4t}$ collapses a stacked power into a single exponent. Going the other way, $3^{4t} = \left(3^4\right)^t = 81^t$ regroups the exponent to expose a new base.

To rewrite an expression with a target base: write the current base as a power of the target, substitute, and multiply exponents. To rewrite $16^t$ with base $2$ , note $16 = 2^4$ , so $16^t = \left(2^4\right)^t = 2^{4t}$ .

Why bother: reading a different rate

In a growth model, the base is a per-period multiplier and the exponent counts periods. Changing the base changes the period you are measuring in. An account that grows by $(1.08)^t$ per year satisfies $(1.08)^t \approx (1.0064)^{12t}$ — and since $12t$ counts months, the rewritten base $1.0064$ shows a monthly growth rate of about $0.64\%$ .

Same account, same values, two readings: $8\%$ per year, or roughly $0.64\%$ per month. Rewriting the base is how you move between them.

Worked examples

Example 1: rewrite $16^t$ with base $2$

Rewrite $y = 16^t$ using base $2$ .

Write

16

as a power of

2

16 = 2^4

Substitute

y = \left(2^4\right)^t

Power of a power multiplies exponents

y = 2^{4t}

Answer: $y = 2^{4t}$

Example 2: rewrite $(1.21)^t$ with base $1.1$

Rewrite $y = (1.21)^t$ using base $1.1$ .

Recognize the square

1.21 = (1.1)^2

Substitute

y = \left((1.1)^2\right)^t

Multiply the exponents

y = (1.1)^{2t}

Answer: $y = (1.1)^{2t}$

Example 3: a fractional exponent

Which expression is equivalent to $\displaystyle 8^{\frac{t}{3}}$ ?

Write

8

as a power of

2

8 = 2^3

Substitute

\displaystyle 8^{\frac{t}{3}} = \left(2^3\right)^{\frac{t}{3}}

Multiply the exponents

3 \cdot \dfrac{t}{3} = t

Answer: $2^t$

Try one yourself

Common questions

Does rewriting the base change the value of the expression?

No. $81^t$ and $3^{4t}$ output identical values for every $t$ . Rewriting only changes the rate you can read directly — the expression stays equivalent.

How do I pick what power to use?

Ask what exponent turns the target base into the current base. For base $3$ and current base $81$ , solve $3^n = 81$ to get $n = 4$ . Then the whole rewrite is $81^t = 3^{4t}$ .

What if the old base is not a whole-number power of the new one?

In Algebra 1 the problems are chosen so it works out — perfect powers like $16 = 2^4$ or recognizable squares like $1.21 = (1.1)^2$ . If nothing fits, double-check for a decimal square or cube before giving up.

Why does the exponent

12t

mean months?

If $t$ counts years, then $12t$ counts twelve steps for every year — months. The exponent always counts how many times the base gets applied, so its scale sets the period of the rate.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Transforming Exponential Expressions

The power of a power rule

Why bother: reading a different rate

Worked examples

Example 1: rewrite $16^t$ with base $2$

Example 2: rewrite $(1.21)^t$ with base $1.1$

Example 3: a fractional exponent

Try one yourself

Common questions

Keep learning

Interpreting Parameters in Context

Writing Exponential Functions

Transformations of Exponential Functions

Transforming Exponential Expressions

The power of a power rule

Why bother: reading a different rate

Worked examples

Example 1: rewrite 16t16^t16t with base 222

Example 2: rewrite (1.21)t(1.21)^t(1.21)t with base 1.11.11.1

Example 3: a fractional exponent

Try one yourself

Common questions

Keep learning

Interpreting Parameters in Context

Writing Exponential Functions

Transformations of Exponential Functions

Example 1: rewrite $16^t$ with base $2$

Example 2: rewrite $(1.21)^t$ with base $1.1$