7-1: Exponential Expressions

Hazel McKenna; Leo Chang; Roxanne Brinkerhoff

CHAPTER 7: EXPONENTIAL AND LOGARITHMIC GROWTH

7-1: Exponential Expressions

Exponents

We use multiplication to represent repeated addition. For example, $8\times3$ is shortcut notation for $3+3+3+3+3+3+3+3$ . Likewise, we use exponents to represent repeated multiplication. For example, $5\times5\times5\times5\times5\times5\times\5$ can be written $5^7$ . The base 5 is multiplied by itself 7 times. 7 is called the exponent. Writing multiple additions or multiple multiplications is tedious and is susceptible to human error, so using multiplication to abbreviate multiple additions, and exponents to abbreviate multiple multiplications makes sense.

Exponents are used regularly to represent area in square units like ft² (e.g., a hotel room is 200 square feet) or m²(e.g., a bedroom is 22 square meters) and volume in cubic units like cm³ (e.g., a box has a volume of 3600 cubic centimeters) or yd³ (e.g., 2 cubic yards of topsoil). So, measuring in two or more dimensions requires the use of exponents on the measurement units.

Any number raised to a power is known as an exponential expression. In section 1-3, we considered powers of 10. We discovered some basic rules that we will see can apply to all numbers.

Exponential Expressions

Explore 1 – Exponential expressions

Suppose a single bacterium is placed in a petri dish. The number of bacteria in the petri dish doubles every hour. The table shows this growth of bacteria.

Time (hours)

0

1

2

3

4

5

Bacteria

1

1×2=2

(2=2¹)

(1×2) x2=4

(4=2²)

(1x2x2) x2=8

(8=2³)

(1x2x2x2) x2=16

(16=2⁴)

(1x2x2x2x2) x2=32

(32=2⁵)

How do we determine the number of bacteria from one hour to the next?

Solution

We multiply the number of bacteria in the first hour by 2.

How many bacteria are there after 6 hours?

Solution

After six hours, the number of bacteria is $2^6 = 64$ (i.e., $2\times2\times2\times2\times2\times2$ ) times the initial number of bacterium, 1.

How many bacteria are there after 1 day?

Solution

Since 1 day = 24 hours, there are $1\times2^{24}=16,777,216$ bacteria.

Is it possible to write the number of bacteria at any time as $2^n$ , where n is the number of hours?

Solution

Yes. The pattern for the number of bacteria is $1,\;2, \;2^2, \;2^3, \;2^4, \;...$ for days 0, 1, 2, 3, 4, … The only day that doesn’t appear to be a power of 2 is day 1 when there is 1 bacterium. But, $2^0=1$ , so the number of bacteria at any time is $2^n$ , where n is the number of hours.

To evaluate an exponential expression using a calculator we use either the carat key $\boxed{\wedge}$ or the exponent key $\boxed{y^x}$ .To evaluate 2²⁴, the input will be $\boxed{2}\;\boxed{\wedge}}\;\boxed{24}\;\boxed{\text{ENTER}}$

Alternatively, the input is $\boxed{2}\;\boxed{y^x}\;\boxed{24}\;\boxed{\text{ENTER}}$

Either way, the result is 16,777,216.

Negative Exponents

Explore 2 – Negative exponents

Knowing that 8 = 2³, 4 = 2², 2 = 2¹, and 1 = 2⁰, follow the pattern to determine what 2^-1 and 2^-2 equal.

Solution

The pattern 8, 4, 2, 1, … divides each number by 2. Consequently it continues to $\frac{1}{2},\;\frac{1}{4},\;\frac{1}{8},\;...$ . At the same time the pattern 2³, 2², 2¹, 2⁰, continues by reducing the exponent each time by 1 to 2^-1, 2^-2, 2^-3, … This means that $2^{-1}=\frac{1}{2}$ and $2^{-2}=\frac{1}{4}$ .

Use the pattern to write $2^{-7}$ as an expression with a positive exponent.

Solution

$2^{-7}=\dfrac{1}{2^7}$

3. Use the pattern to write $2^{-n}$ as an expression with a positive exponent.

Solution

From the pattern, $2^{-7}=\dfrac{1}{2^7}$ . So, $2^{-n}=\dfrac{1}{2^n}$

Expand your expression to write $a^{-n}$ for any non-zero value $a$ , as an expression with a positive exponent.

Solution

$a^{-n}=\dfrac{1}{a^n}$

The Product and Quotient Rules

Explore 3 – The product rule

Is it true that 2 × 2 × 2 × 2 × 2 is the same as (2 × 2) × (2 × 2 × 2)? Explain your reasoning.

Solution

Yes, they are the same.

$\begin{aligned}2 \times 2 \times 2 \times 2 \times 2 &\\= 4\times2\times2\times2 &\\=8\times2\times2 &\\=16\times2 &\\=32\end{aligned}$ and $\begin{aligned}(2 \times 2) \times (2 \times 2 \times 2) &\\= 4 \times (4\times2)&\\=4\times8&\\ = 32\end{aligned}$

There are two sets of parentheses that change the grouping of the multiplication but do not change the value of the multiplication of the five 2s. This is an example of the associative property of multiplication.

Considering your answer to #1, is it true that $2^2\times2^3 = 2^5$ . Explain your reasoning.

Solution

Yes. #1 shows us that $(2\times2)\times(2\times2\times2)=2 \times 2 \times 2 \times 2 \times 2$ . Since $2\times2=2^2$ , $2\times2\times2=2^3$ , and $2 \times 2 \times 2 \times 2 \times 2=2^5$ , then $2^2\times2^3=2^5$ .

Since $2^2\times2^3=2^5$ , is there an easy rule to multiply exponential expressions with the same base?

Solution

Yes. Since 2 + 3 = 5, we can keep the common base and add the exponents. This is called the product rule of exponents: $a^m\times a^n=a^{m+n}$

Use the product rule to simplify $2^8\times2^{12}$

Solution

Since 2⁸ and 2¹² have the same base 2, we keep the common base and add the exponents: $2^8\times2^{12}=2^{8+12}=2^{20}$

Use the product rule to simplify $5^5\times5^4\times5^8$

Solution

Since 5⁵, 5⁴, and 5⁸ all have the same base, 5, we keep the common base and add the exponents: $5^5\times5^4\times5^8=5^{5+4+8}=5^{17}$

Explore 4 – The quotient rule

Simplify $\dfrac{5^{7}}{5^{4}}$ by simplifying the fraction.

Solution

$\dfrac{5^{7}}{5^{4}}=\dfrac{\cancel{5}\times\cancel{5}\times\cancel{5}\times\cancel{5}\times5\times5\times5}{\cancel{5}\times\cancel{5}\times\cancel{5}\times\cancel{5}}=\dfrac{5^3}{1}=5^3$ .

Use multiplication and the fact that $a^{-n}=\dfrac{1}{a^n}$ to show that $\dfrac{7^8}{7^5}=7^{3}$

Solution

$\dfrac{7^8}{7^5}=7^{3}=7^8\cdot 7^{-5}=7^{8+(-5)}=7^3$

Since $\dfrac{5^{7}}{5^{4}}=5^3$ and $\dfrac{7^8}{7^5}=7^{3}$ , is there an easy rule to simplify the division?

Solution

For exponential expressions with the same base, we can divide the expressions by keeping the common base and subtracting the exponents.

Use your rule to simplify: $\dfrac{5^{12}}{5^9}$

Solution

With a common base of 5, we subtract the exponents: $\dfrac{5^{12}}{5^9}=5^{12-9}=5^3$

Reflect Icon

Explain the product rule $a^m\times a^n=a^{m+n}$ and the quotient rule $\dfrac{a^m}{a^n}=a^{m-n}$ . In particular, explain why it is $m+n$ and $m-n$ respectively in the rules.

Show/Hide Answer

It is $m+n$ in the product rule because $a$ is multiplied $m$ times and then multiplied $n$ times. Therefore, $a$ is multiplied $m+n$ times. It is $m-n$ in the quotient rule because $a$ is multiplied $m$ times, and then divided $n$ times, which means that $a$ is multiplied $m-n$ times.

The Product/Quotient to a Power Rule

Explore 5 – The product to a power rule

Evaluate the expression $(2\times3)^5$ following the order of operations.

Solution

The order of operation tells us to first complete the multiplication inside the parentheses: $(2\times3)^5=6^5$

Then we evaluate the exponent: $6^5=7776$

Evaluate the expression $2^5\times3^5$ following the order of operations.

Solution

First we evaluate the exponents: $2^5\times3^5=32\times243$

Then we multiply: $32\times243=7776$

What do the solutions to 1. and 2. tell us?

Solution

Since the answers are the same (i.e., 7776), $2^5\times3^5=(2\times3)^5$ .

Write a rule in English that describes how to simplify $(4x)^3$ , then use it to complete the simplification.

Solution

A product to a power, like $(4x)^3$ , can be simplified by applying the exponent to both factors, then multiplying: $(4x)^3=4^3\cdot x^3=64x^3$

Complete the algebraic rule: $(ab)^n=\;\;\;\;\;\;\;\;$

Solution

$(ab)^n=a^n\cdot b^n$

Explore 6 – The quotient to a power rule

Write the expression $\left(\dfrac{2}{3}\right)^4$ as a product of fractions, then simplify.

Solution

$\left(\dfrac{2}{3}\right)^4=\dfrac{2}{3}\cdot\dfrac{2}{3}\cdot\dfrac{2}{3}\cdot\dfrac{2}{3}=\dfrac{2\cdot 2\cdot 2\cdot 2}{3\cdot 3\cdot 3\cdot 3}=\dfrac{2^4}{3^4}$

Write the expression $\left(\dfrac{3}{4}\right)^5$ as a product of fractions, then simplify.

Solution

$\left(\dfrac{3}{4}\right)^5=\dfrac{3}{4}\cdot\dfrac{3}{4}\cdot\dfrac{3}{4}\cdot\dfrac{3}{4}\cdot\dfrac{3}{4}=\dfrac{3\cdot 3\cdot 3\cdot 3\cdot 3}{4\cdot 4\cdot 4\cdot 4\cdot 4}=\dfrac{3^5}{4^5}$

Write the expression $\left(\dfrac{a}{b}\right)^5$ as a product of fractions, then simplify.

Solution

$\left(\dfrac{a}{b}\right)^5=\dfrac{a}{b}\cdot\dfrac{a}{b}\cdot\dfrac{a}{b}\cdot\dfrac{a}{b}\cdot\dfrac{a}{b}=\dfrac{a\cdot a\cdot a\cdot a\cdot a}{b\cdot b\cdot b\cdot b\cdot b}=\dfrac{a^5}{b^5}$

Complete the algebraic rule: $\left(\dfrac{a}{b}\right)^n=$

Solution

$\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$

Does your rule work for all values of $a$ and $b$ ?

Solution

No. Since we cannot divide by zero, $b\ne 0$

Reflect Icon

Explain the product to a power rule $(a\times b)^n=a^n\times b^n$ and the quotient to a power rule $(\frac{a}{b})^n=\frac{a^n}{b^n}$ . In particular, explain why the exponent n is applied to to a and b.

Show/Hide Answer

For the product to a power rule, the reason $n$ may be applied to $a$ and $b$ is that the product $(a\times b)$ is multiplied together $n$ times. The associative (i.e., $(ab)c=a(bc)$ ) and commutative (i.e., $ab = ba$ ) properties of multiplication can be applied to get a result that is equal to $a^n\times b^n$ . In other words, we rearrange the $ab$ s so that all the $a$ s are together and all the $b$ s are together.

For the quotient to a power rule, the reason $n$ may be applied to $a$ and $b$ is that the fraction $\dfrac{a}{b}$ is multiplied together $n$ times. We multiply all the numerators and put the result in the numerator and multiply all the denominators and put the result in the denominator. There are $n$ terms of $a$ ‘s multiplied together in the numerator and $n$ terms of $b$ in the denominator multiplied together. Therefore, the result is $\dfrac{a^n}{b^n}$ .

Power to a Power Rule

Explore 7 – Power to a power rule

To understand the exponential expression $(2^6)^4$ , Nina starts with the exponent 4, and comes up with the equivalent expression $2^6\times2^6\times2^6\times2^6$ .

Explain why there are four $2^6$ terms.

Solution

The meaning of the exponent 4 means there are four of the base $2^6$ that are multiplied together. Therefore, $(2^6)^4=2^6\times2^6\times2^6\times2^6$ .

How many 2’s are there in total in the multiplication?

Solution

Based on the product rule of exponents, $2^6\times2^6\times2^6\times2^6=2^{6+6+6+6}=2^{24}$ . Therefore, there are 24 twos multiplied together.

Nina explains that the total number of 2’s is 6 x 4 = 24. Why can Nina multiply the two powers?

Solution

It is because there are four $2^6$ multiplied together. The total number of 2 that are multiplied together is $6+6+6+6=4\times6$ .

Write a rule that simplifies a power raised to a power, then use it to simplify $\left(3^5\right)^7$ .

Solution

$\left(a^m\right)^n=a^{mn}$ $\left(3^5\right)^7=3^{5\times 7}=3^{35}$

Explore 8 – Power to a power rule

A population of bacteria doubles six times in one hour.

Jonas says the population after 4 hours will be $\left(2^6\right)^4$ times the initial population. Is he correct? Explain your reasoning.

Solution

Doubling six times in one hour means the population doubles every 10 minutes. If the initial population is $p$ then after the first 10 minutes there will be $2p$ ; then after the second 10 minutes there are $2\cdot2p=2^2p$ ; then after the third 10 minutes there are $2\cdot2^2p=2^3p$ ; …so after the sixth 10 minutes (i.e., one hour) there will be $\left(2^6\right)p$ bacteria.

Consequently, after 4 hours there will be $(2^6)\times(2^6)\times(2^6)\times(2^6)p=\left(2^6\right)^4p$ .

Simplify $\left(2^6\right)^4$ without using any exponents.

Solution

$(2\times2\times2\times2\times2\times2)\times(2\times2\times2\times2\times2\times2)\times(2\times2\times2\times2\times2\times2)\times(2\times2\times2\times2\times2\times2)=2^{24}$

Simplify $\left(2^6\right)^4$ using the product rule of exponents.

Solution

$(2^6)\times(2^6)\times(2^6)\times(2^6)=2^{6+6+6+6}=2^{24}$ .

Simplify $\left(2^6\right)^4$ using the power to a power rule.

Solution

$(2^6)^4=2^{4\times6}=2^{24}$ .

Reflect Icon

Explain the power to a power rule $(a^m)^n=a^{m\times n}$ . In particular, explain why it is $m\times n$ .

Show/Hide Answer

Since there are $n$ of $a^m$ multiplied together, it shows $a^m\times a^m\times...\times a^m = a^{m+m+...+m}$ . Since there are $n$ of $m$ , the sum of the addition of $m$ ‘s is $n\times m$ . Therefore, $(a^m)^n=a^{m\times n}$ .

Rules of Exponents

Exponent of 1:	Any number to the power 1 equals the number.	$a^1=a$
Zero Exponent:	Any non-zero number to the power zero equals 1.	$a^0=1\;\;a\ne0$
Product Rule	Two or more exponential expressions with the same base can be multiplied by keeping the common base and adding the exponents.	$a^m\times a^n=a^{m+n}$
Quotient Rule	Two exponential expressions with the same base can be divided by keeping the common base and subtracting the exponents.	$a^m\div a^n=a^{m-n}$
Negative Exponents	When an exponential expression has a negative exponent the number is a fraction of a whole.	$a^{-n}=1\div a^n\;\;a\ne0$
Product to a Power Rule	A product can be raised to a power by multiplying each factor raised to the power.	$(ab)^n=a^n\,b^n$
Quotient to a Power Rule	A quotient raised to a power can be split into the numerator raised to the power divided by the denominator raised to the power.	$\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$
Power to a Power Rule	To raise a power to a power, keep the base and multiply the exponents.	$(a^m)^n=a^{mn}$

Example

The number of people waiting for a train that is running late doubles every 15 minutes. a) Complete the table to show how the number of people has grown over time. b) Write an expression that shows the growth after an hour. c) The next train is almost due, but the late train has not yet arrived. Consequently, the number of people waiting for the train triples every 15 minutes after the first hour. Complete the table for the second hour. c) Simplify the exponential form after 2 hours. d) If there were originally three people waiting for the train, how many people are waiting on the train after 2 hours? e) If each train carriage holds 630 passengers, how many carriages will be needed to accommodate the waiting passengers after 2 hours?

Time (hours)	0	0.25	0.5	0.75	1	1.25	1.5	1.75	2
People	$p$	$2p$
Exponential form		$2^1p$

Show/Hide Answer

Time (hours) 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

People $p$ $2p$ $4p$ $8p$ $16p$ $48p$ $144p$ $432p$ $1296p$

Exponential form $2^0p$ $2^1p$ $2^2p$ $2^3p$ $2^4p$ $3\cdot 2^4p$ $3^2\cdot2^4p$ $3^3\cdot2^4p$ $3^4\cdot2^4p$

c) $3^4\cdot2^4p=6^4p$ d) $6^4(3)=3888$ e) 3888/630 = 6.2; 7 carriages will be required.

Practice Exercises

Use the rules of exponents to simplify. Leave your answers with only positive exponents.

$(2^4)^3\cdot 2^5\cdot 2^{-7}$
$(2x)^6\cdot x^5$
$(5x^2)^3$
$(2^3)^{-4}$
$(7x^3)^2\cdot (5x^{-2})^2$
$\dfrac{7x^{12}}{x^{-8}}$
Nathan says $(2\times3)^5$ is the same as $2\times3^5$ . Is he correct? Explain your reasoning.

Show/Hide Answer

$2^{10}$
$2^6x^{11}$
$5^3x^6$
$\dfrac{1}{2^{12}}$
$35^2x^2$
$7x^{20}$
No. $(2\times3)^5=6^5=7776$ , while $2\times3^5=2\times 243=486$ .

In this section, we will take what we have learned and apply the concepts to new situations.

Perspectives

Jean is moving apartments and doesn’t want to take all of her junk with her when she moves. She has decided to throw away 1 item the first day, then double that number of items she discards each day over the next week. a) Create a table to show how many items will be discarded each day. b) How many items will she discard on day $n$ ?
A population of rabbits doubles every 10 days. a) Create a table to show how many rabbits there are in the population over time if there were initially $p$ rabbits in the population. b) Write an algebraic expression showing how many rabbits there are after $10n$ days. c) If left unchecked, how many rabbits will there be after 360 days?
Adriel deposits $5000 into his bank account. The account balance increases to 1.015 times the balance every year. What is the account balance after 10 years? Show your work using exponents.
The population of a species doubles every month from January to April, and then reduces by half every month for the following two months. a) Write the population increase-decrease pattern in a mathematical sentence without using exponents. b) Write the population increase-decrease pattern in a mathematical sentence using exponents. c) Simplify the expression using the rules of exponents.
A video clip goes viral. The number of people who view the clip doubles every 15 minutes for the first six hours, and then triples every 12 hours for the next 4 days. Write the population growth as a mathematical sentence using exponents.