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Manipulating Exponential Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Equivalent forms of exponential expressions

Why rewrite exponential expressions?

Exponential expressions can often be rewritten in equivalent forms using the properties of exponents
Different forms of the same expression can reveal different information
- E.g. one form might make the base clearer
- another might show a transformation more explicitly
Being confident with these properties is essential for
- solving equations
- simplifying expressions
- and working with exponential models

What is the product property for exponents?

The product property states that
- $b^{m} \cdot b^{n} = b^{(m + n)}$
  - I.e. when you multiply powers with the same base, you add the exponents
- E.g. $2^{3} \cdot 2^{5} = 2^{8}$ , and $5^{x} \cdot 5^{2} = 5^{(x + 2)}$
With reference to graphs this means that
- Every horizontal translation of an exponential function
- is equivalent to a vertical dilation
Consider $f (x) = b^{(x + k)}$
- I.e. a horizontal translation of $b^{x}$ by $k$ units to the left
Using the product property, $f (x) = b^{x} \cdot b^{k} = (b^{k}) \cdot b^{x}$
- Since $b^{k}$ is a constant, this is just the function $b^{x}$ multiplied by the constant $b^{k}$
- which is a vertical stretch by a factor of $b^{k}$
So shifting the graph of $b^{x}$ horizontally by $k$ units is the same as multiplying $b^{x}$ by $b^{k}$
E.g. $3^{(x + 2)} = 3^{2} \cdot 3^{x} = 9 \cdot 3^{x}$
- Shifting $3^{x}$ left by 2 units is equivalent to vertically stretching by a factor of 9

What is the power property for exponents?

The power property states that
- $(b^{m})^{n} = b^{m n}$
  - I.e. when you raise a power to another power, you multiply the exponents
- E.g. $(2^{3})^{4} = 2^{12}$ , and $(5^{2})^{x} = 5^{2 x}$
With reference to graphs this means that
- Every horizontal dilation of an exponential function
- is equivalent to a change of the base
Consider $f (x) = b^{(c x)}$
- I.e. a horizontal dilation of $b^{x}$ by a factor of $\frac{1}{c}$
- where $c \neq 0$
Using the power property, $f (x) = (b^{c})^{x}$
- Since $b^{c}$ is a constant, this is just another exponential function with the new base $b^{c}$

So horizontally compressing or stretching the graph of $b^{x}$ changes the base
E.g. $3^{(2 x)} = (3^{2})^{x} = 9^{x}$
- Horizontally dilating $3^{x}$ by a factor of $\frac{1}{2}$ is equivalent to changing the base to 9

What is the negative exponent property?

The negative exponent property states that
- $b^{- n} = \frac{1}{b^{n}}$
  - I.e., a negative exponent means the reciprocal of the positive power
- E.g. $2^{- 3} = \frac{1}{2^{3}} = \frac{1}{8}$ , and $5^{- x} = \frac{1}{5^{x}} = {(\frac{1}{5})}^{x}$
This property is useful for converting between growth and decay forms
E.g. $2^{- x} = {(\frac{1}{2})}^{x}$
- A negative exponent on a base greater than 1
- is equivalent to a positive exponent on a base between 0 and 1

What are unit fraction exponents?

The value of an exponential expression involving a unit fraction exponent represents a root
- I.e. $b^{1 / k}$ where $k$ is a natural number
  - is the $k$ th root of $b$ (when it exists)
- $b^{1 / 2} = \sqrt{b}$
- $b^{1 / 3} = \sqrt[3]{b}$
- $b^{1 / k} = \sqrt[k]{b}$
This connects fractional exponents to roots
- and is often used when simplifying exponential expressions
Combined with the power property, this gives
- $b^{m / k} = {(b^{m})}^{1 / k} = \sqrt[k]{b^{m}}$
  - or $b^{m / k} = {(b^{1 / k})}^{m} = {(\sqrt[k]{b})}^{m}$
- E.g. $8^{2 / 3} = \sqrt[3]{8^{2}} = \sqrt[3]{64} = 4$
  - or alternatively $8^{2 / 3} = {(\sqrt[3]{8})}^{2} = {(2)}^{2} = 4$

How can I change the base of an exponential expression?

To convert an exponential expression to an equivalent expression with a different base
- express the original base as a power of the new base
- then apply the power property

E.g. to rewrite $4^{x}$ in terms of base 2
- since $4 = 2^{2}$ , you get $4^{x} = (2^{2})^{x}$
- then the power property gives you $4^{x} = 2^{2 x}$

Examiner Tips and Tricks

When rewriting exponential expressions in equivalent forms, the key is to apply the exponent properties step by step.

Avoid trying to do too many steps at once, as errors are easily introduced
Be careful with expressions like $a \cdot b^{(x + k)}$
- Use the product property to separate the constant part before simplifying
  - $a \cdot b^{(x + k)} = a \cdot b^{k} \cdot b^{x} = (a \cdot b^{k}) \cdot b^{x}$

Worked Example

The function $f$ is given by $f (x) = 4 \cdot 9^{(x + 1)}$ . Which of the following is an equivalent form for $f (x)$ ?

(A) $f (x) = 2 \cdot 3^{(\frac{x + 1}{2})}$

(B) $f (x) = 2 \cdot 3^{(2 x + 2)}$

(D) $f (x) = 36 \cdot 3^{(2 x)}$

Answer:

All of the answer options have 3 raised to a power instead of 9

So start by rewriting the $9^{(x + 1)}$ part of the original expression

One approach is to start by writing $9 = 3^{2}$ and then using laws of indices

$\begin{array}{rcl} 9^{(x + 1)} & = & {(3^{2})}^{(x + 1)} \\ = & 3^{2 (x + 1)} \\ = & 3^{(2 x + 2)} \end{array}$

That looks a bit like option B, but when you put it back into the original expression you just get $4 \cdot 3^{(2 x + 2)}$ , not $2 \cdot 3^{(2 x + 2)}$

Another approach is to start by using laws of indices to rewrite $9^{(x + 1)}$ first

and only then to bring in $9 = 3^{2}$

$\begin{array}{rcl} 9^{(x + 1)} & = & 9^{x} \cdot 9^{1} \\ = & 9 \cdot 9^{x} \\ = & 9 \cdot {(3^{2})}^{x} \\ = & 9 \cdot 3^{(2 x)} \end{array}$

Substituting that back into the original expression gives

$\begin{array}{rcl} 4 \cdot 9^{(x + 1)} & = & 4 \cdot 9 \cdot 3^{(2 x)} \\ = & 36 \cdot 3^{(2 x)} \end{array}$

So option D is an equivalent expression for $f (x)$

(D) $f (x) = 36 \cdot 3^{(2 x)}$

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