Contents

Simplify and Use Square Roots

Learning Objectives

By the end of this section, you will be able to:

Simplify expressions with square roots
Estimate square roots
Approximate square roots
Simplify variable expressions with square roots
Use square roots in applications

Be Prepared 5.7

Before you get started, take this readiness quiz.

Simplify: ${(−9)}^{2} . {(−9)}^{2} .$
If you missed this problem, review Example 3.52.
Round $3.846 3.846$ to the nearest hundredth.
If you missed this problem, review Example 5.9.
Evaluate $12 d 12 d$ for $d = 80 . d = 80 .$
If you missed this problem, review Example 2.14.

Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number $n n$ is multiplied by itself, we can write this as $n^{2}, n^{2},$ which we read aloud as $“ n squared.” “ n$ For example, $8^{2} 8^{2}$ is read as $“8 squared.” “8$

We call $6464$ the square of $88$ because $8^{2} = 64 . 8^{2} = 64 .$ Similarly, $121121$ is the square of $11, 11,$ because $11^{2} = 121 . 11^{2} = 121 .$

Square of a Number

If $n^{2} = m, n^{2} = m,$ then $m m$ is the square of $n . n .$

Modeling Squares

Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine.

A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.

This is why we say that the square of three is nine.

3^{2} = 9 3^{2} = 9

The number $99$ is called a perfect square because it is the square of a whole number.

Manipulative Mathematics

Doing the Manipulative Mathematics activity Square Numbers will help you develop a better understanding of perfect square numbers

The chart shows the squares of the counting numbers $11$ through $15 . 15 .$ You can refer to it to help you identify the perfect squares.

Perfect Squares

A perfect square is the square of a whole number.

What happens when you square a negative number?

\begin{matrix} {(−8)}^{2} & = (−8) (−8) \\ = 64 \end{matrix} \begin{matrix} {(−8)}^{2} & = (−8) (−8) \\ = 64 \end{matrix}

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.

The chart shows the squares of the negative integers from $−1 −1$ to $−15 . −15 .$

Did you notice that these squares are the same as the squares of the positive numbers?

Square Roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because $10^{2} = 100, 10^{2} = 100,$ we say $100100$ is the square of $10 . 10 .$ We can also say that $1010$ is a square root of $100 . 100 .$

Square Root of a Number

A number whose square is $m m$ is called a square root of $m . m .$

If $n^{2} = m, n^{2} = m,$ then $n n$ is a square root of $m . m .$

Notice ${(−10)}^{2} = 100 {(−10)}^{2} = 100$ also, so $−10 −10$ is also a square root of $100 . 100 .$ Therefore, both $1010$ and $−10 −10$ are square roots of $100 . 100 .$

So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, $\sqrt{}, \sqrt{},$ stands for the positive square root. The positive square root is also called the principal square root.

Square Root Notation

$\sqrt{m} \sqrt{m}$ is read as “the square root of $m .” m .”$

$If m = n^{2}, then \sqrt{m} = n for n \geq 0 . If$

We can also use the radical sign for the square root of zero. Because $0^{2} = 0, \sqrt{0} = 0 . 0^{2} = 0, \sqrt{0} = 0 .$ Notice that zero has only one square root.

The chart shows the square roots of the first $1515$ perfect square numbers.

Example 5.69

Simplify: ⓐ $\sqrt{25}$ ⓑ $\sqrt{121} .$

Show solution

Solution

ⓐ
	$\sqrt{25} \sqrt{25}$
Since $5^{2} = 25 5^{2} = 25$	$55$

ⓑ
	$\sqrt{121} \sqrt{121}$
Since $11^{2} = 121 11^{2} = 121$	$1111$

Try It 5.137

Simplify: ⓐ $\sqrt{36}$ ⓑ $\sqrt{169} .$

Try It 5.138

Simplify: ⓐ $\sqrt{16}$ ⓑ $\sqrt{196} .$

Every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{100} = 10 . \sqrt{100} = 10 .$ If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $- \sqrt{100} = −10 . - \sqrt{100} = −10 .$

Example 5.70

Simplify. ⓐ $- \sqrt{9}$ ⓑ $- \sqrt{144 .}$

Show solution

Solution

ⓐ
	$- \sqrt{9} - \sqrt{9}$
The negative is in front of the radical sign.	$- 3 - 3$

ⓑ
	$- \sqrt{144} - \sqrt{144}$
The negative is in front of the radical sign.	$- 12 - 12$

Try It 5.139

Simplify: ⓐ $- \sqrt{4}$ ⓑ $- \sqrt{225} .$

Try It 5.140

Simplify: ⓐ $- \sqrt{81}$ ⓑ $- \sqrt{64} .$

Square Root of a Negative Number

Can we simplify $\sqrt{−25} ? \sqrt{−25} ?$ Is there a number whose square is $−25 ? −25 ?$

{()}^{2} = −25 ? (^{2} = −25 ?

None of the numbers that we have dealt with so far have a square that is $−25 . −25 .$ Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to $\sqrt{−25} . \sqrt{−25} .$ If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Example 5.71

Simplify: ⓐ $\sqrt{−169}$ ⓑ $- \sqrt{121} .$

Show solution

Solution

ⓐ There is no real number whose square is $−169 . −169 .$ Therefore, $\sqrt{−169} \sqrt{−169}$ is not a real number.

ⓑ The negative is in front of the radical sign, so we find the opposite of the square root of $121 . 121 .$

	$- \sqrt{121} - \sqrt{121}$
The negative is in front of the radical.	$- 11 - 11$

Try It 5.141

Simplify: ⓐ $\sqrt{−196}$ ⓑ $- \sqrt{81} .$

Try It 5.142

Simplify: ⓐ $\sqrt{−49}$ ⓑ $- \sqrt{121} .$

Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Example 5.72

Simplify: ⓐ $\sqrt{25} + \sqrt{144}$ ⓑ $\sqrt{25 + 144} .$

Show solution

Solution

ⓐ Use the order of operations.
	$\sqrt{25} + \sqrt{144} \sqrt{25} + \sqrt{144}$
Simplify each radical.	$5 + 12 5 + 12$
Add.	$1717$

ⓑ Use the order of operations.
	$\sqrt{25 + 144} \sqrt{25 + 144}$
Add under the radical sign.	$\sqrt{169} \sqrt{169}$
Simplify.	$1313$

Try It 5.143

Simplify: ⓐ $\sqrt{9} + \sqrt{16}$ ⓑ $\sqrt{9 + 16} .$

Try It 5.144

Simplify: ⓐ $\sqrt{64 + 225}$ ⓑ $\sqrt{64} + \sqrt{225} .$

Notice the different answers in parts ⓐ and ⓑ of Example 5.72. It is important to follow the order of operations correctly. In ⓐ, we took each square root first and then added them. In ⓑ, we added under the radical sign first and then found the square root.

Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled “Square root” and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.

We might conclude that the square roots of numbers between $44$ and $99$ will be between $22$ and $3, 3,$ and they will not be whole numbers. Based on the pattern in the table above, we could say that $\sqrt{5} \sqrt{5}$ is between $22$ and $3 . 3 .$ Using inequality symbols, we write

2 < \sqrt{5} < 3 2 < \sqrt{5} < 3

Example 5.73

Estimate $\sqrt{60} \sqrt{60}$ between two consecutive whole numbers.

Show solution

Solution

Think of the perfect squares closest to $60 . 60 .$ Make a small table of these perfect squares and their squares roots.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 36, 49, 64, and 81. There is a balloon coming out of the table between 49 and 64 that says 60. The second column is labeled “Square root” and contains the values: 6, 7, 8, and 9. There is a balloon coming out of the table between 7 and 8 that says square root of 60.

$Locate 60 between two consecutive perfect squares. Locate 60 between two consecutive perfect squares.$	$49 < 60 < 64 49 < 60 < 64$
$\sqrt{60} is between their square roots. \sqrt{60}$	$7 < \sqrt{60} < 8 7 < \sqrt{60} < 8$

Try It 5.145

Estimate $\sqrt{38} \sqrt{38}$ between two consecutive whole numbers.

Try It 5.146

Estimate $\sqrt{84} \sqrt{84}$ between two consecutive whole numbers.

Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the $\sqrt{} \sqrt{}$ or $\sqrt{x} \sqrt{x}$ key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is $\approx \approx$ and it is read approximately.

Suppose your calculator has a $10-digit 10-digit$ display. Using it to find the square root of $55$ will give $2.236067977 . 2.236067977 .$ This is the approximate square root of $5 . 5 .$ When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

\sqrt{5} \approx 2.236067978 \sqrt{5} \approx 2.236067978

You will seldom use this many digits for applications in algebra. So, if you wanted to round $\sqrt{5} \sqrt{5}$ to two decimal places, you would write

\sqrt{5} \approx 2.24 \sqrt{5} \approx 2.24

How do we know these values are approximations and not the exact values? Look at what happens when we square them.

\begin{matrix} {2.236067978}^{2} & = & 5.000000002 \\ {2.24}^{2} & = & 5.0176 \end{matrix} \begin{matrix} {2.236067978}^{2} & = & 5.000000002 \\ {2.24}^{2} & = & 5.0176 \end{matrix}

The squares are close, but not exactly equal, to $5 . 5 .$

Example 5.74

Round $\sqrt{17} \sqrt{17}$ to two decimal places using a calculator.

Show solution

Solution

	$\sqrt{17} \sqrt{17}$
Use the calculator square root key.	$4.123105626 4.123105626$
Round to two decimal places.	$4.12 4.12$
	$\sqrt{17} \approx 4.12 \sqrt{17} \approx 4.12$

Try It 5.147

Round $\sqrt{11} \sqrt{11}$ to two decimal places.

Try It 5.148

Round $\sqrt{13} \sqrt{13}$ to two decimal places.

Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider $\sqrt{9 x^{2}}, \sqrt{9 x^{2}},$ where $x \geq 0 . x \geq 0 .$ Can you think of an expression whose square is $9 x^{2} ? 9 x^{2} ?$

\begin{matrix} {(?)}^{2} & = & 9 x^{2} \\ {(3 x)}^{2} & = & 9 x^{2} so \sqrt{9 x^{2}} = 3 x \end{matrix} \begin{matrix} {(?)}^{2} & = & 9 x^{2} \\ {(3 x)}^{2} & = & 9 x^{2} \end{matrix}

When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

Example 5.75

Simplify: $\sqrt{x^{2}} . \sqrt{x^{2}} .$

Show solution

Solution

Think about what we would have to square to get $x^{2} x^{2}$ . Algebraically, ${(?)}^{2} = x^{2} {(?)}^{2} = x^{2}$

	$\sqrt{x^{2}} \sqrt{x^{2}}$
Since ${(x)}^{2} = x^{2} {(x)}^{2} = x^{2}$	$x x$

Try It 5.149

Simplify: $\sqrt{y^{2}} . \sqrt{y^{2}} .$

Try It 5.150

Simplify: $\sqrt{m^{2}} . \sqrt{m^{2}} .$

Example 5.76

Simplify: $\sqrt{16 x^{2}} . \sqrt{16 x^{2}} .$

Show solution

Solution

	$\sqrt{16 x^{2}} \sqrt{16 x^{2}}$
$Since {(4 x)}^{2} = 16 x^{2} Since$	$4 x 4 x$

Try It 5.151

Simplify: $\sqrt{64 x^{2}} . \sqrt{64 x^{2}} .$

Try It 5.152

Simplify: $\sqrt{169 y^{2}} . \sqrt{169 y^{2}} .$

Example 5.77

Simplify: $- \sqrt{81 y^{2}} . - \sqrt{81 y^{2}} .$

Show solution

Solution

	$- \sqrt{81 y^{2}} - \sqrt{81 y^{2}}$
$Since {(9 y)}^{2} = 81 y^{2} Since$	$- 9 y - 9 y$

Try It 5.153

Simplify: $- \sqrt{121 y^{2}} . - \sqrt{121 y^{2}} .$

Try It 5.154

Simplify: $- \sqrt{100 p^{2}} . - \sqrt{100 p^{2}} .$

Example 5.78

Simplify: $\sqrt{36 x^{2} y^{2}} . \sqrt{36 x^{2} y^{2}} .$

Show solution

Solution

	$\sqrt{36 x^{2} y^{2}} \sqrt{36 x^{2} y^{2}}$
$Since {(6 x y)}^{2} = 36 x^{2} y^{2} Since$	$6 x y 6 x y$

Try It 5.155

Simplify: $\sqrt{100 a^{2} b^{2}} . \sqrt{100 a^{2} b^{2}} .$

Try It 5.156

Simplify: $\sqrt{225 m^{2} n^{2}} . \sqrt{225 m^{2} n^{2}} .$

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

How To

Use a strategy for applications with square roots.

Step 1. Identify what you are asked to find.
Step 2. Write a phrase that gives the information to find it.
Step 3. Translate the phrase to an expression.
Step 4. Simplify the expression.
Step 5. Write a complete sentence that answers the question.

Square Roots and Area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is $A A$ square units, the length of a side is $\sqrt{A} \sqrt{A}$ units. See Table 5.7.

Area (square units)	Length of side (units)
$99$	$\sqrt{9} = 3 \sqrt{9} = 3$
$144144$	$\sqrt{144} = 12 \sqrt{144} = 12$
$A A$	$\sqrt{A} \sqrt{A}$

Table 5.7

Example 5.79

Mike and Lychelle want to make a square patio. They have enough concrete for an area of $200200$ square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

Show solution

Solution

We know the area of the square is $200200$ square feet and want to find the length of the side. If the area of the square is $A A$ square units, the length of a side is $\sqrt{A} \sqrt{A}$ units.

What are you asked to find?	The length of each side of a square patio
Write a phrase.	The length of a side
Translate to an expression.	$\sqrt{A} \sqrt{A}$
Evaluate $\sqrt{A} \sqrt{A}$ when $A = 200 A = 200$ .	$\sqrt{200} \sqrt{200}$
Use your calculator.	$14.142135 ... 14.142135 ...$
Round to one decimal place.	$14.1 feet 14.1 feet$
Write a sentence.	Each side of the patio should be $14.1 14.1$ feet.

Try It 5.157

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of $370370$ square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?

Try It 5.158

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of $27042704$ square centimeters. How long can a side of his mosaic be?

Square Roots and Gravity

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of $h h$ feet, the time in seconds it will take to reach the ground is found by evaluating the expression $\frac{\sqrt{h}}{4} . \frac{\sqrt{h}}{4} .$ For example, if an object is dropped from a height of $6464$ feet, we can find the time it takes to reach the ground by evaluating $\frac{\sqrt{64}}{4} . \frac{\sqrt{64}}{4} .$

	$\frac{\sqrt{64}}{4} \frac{\sqrt{64}}{4}$
Take the square root of 64.	$\frac{8}{4} \frac{8}{4}$
Simplify the fraction.	$22$

It would take $22$ seconds for an object dropped from a height of $6464$ feet to reach the ground.

Example 5.80

Christy dropped her sunglasses from a bridge $400400$ feet above a river. How many seconds does it take for the sunglasses to reach the river?

Show solution

Solution

What are you asked to find?	The number of seconds it takes for the sunglasses to reach the river
Write a phrase.	The time it will take to reach the river
Translate to an expression.	$\frac{\sqrt{h}}{4} \frac{\sqrt{h}}{4}$
Evaluate $\frac{\sqrt{h}}{4} \frac{\sqrt{h}}{4}$ when $h = 400 h = 400$ .	$\frac{\sqrt{400}}{4} \frac{\sqrt{400}}{4}$
Find the square root of 400.	$\frac{20}{4} \frac{20}{4}$
Simplify.	$55$
Write a sentence.	It will take 5 seconds for the sunglasses to reach the river.

Try It 5.159

A helicopter drops a rescue package from a height of $12961296$ feet. How many seconds does it take for the package to reach the ground?

Try It 5.160

A window washer drops a squeegee from a platform $196196$ feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?

Square Roots and Accident Investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is $d d$ feet, then the speed of the car can be found by evaluating $\sqrt{24 d} . \sqrt{24 d} .$

Example 5.81

After a car accident, the skid marks for one car measured $190190$ feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?

Show solution

Solution

What are you asked to find?	The speed of the car before the brakes were applied
Write a phrase.	The speed of the car
Translate to an expression.	$\sqrt{24 d} \sqrt{24 d}$
Evaluate $\sqrt{24 d}$ when $d = 190 .$	$\sqrt{24 \cdot 190} \sqrt{24 \cdot 190}$
Multiply.	$\sqrt{4,560} \sqrt{4,560}$
Use your calculator.	$67.527772 ... 67.527772 ...$
Round to tenths.	$67.5 67.5$
Write a sentence.	The speed of the car was approximately 67.5 miles per hour.

Try It 5.161

An accident investigator measured the skid marks of a car and found their length was $7676$ feet. To the nearest tenth, what was the speed of the car before the brakes were applied?

Try It 5.162

The skid marks of a vehicle involved in an accident were $122122$ feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?

Links To Literacy

The Links to Literacy activity "Sea Squares" will provide you with another view of the topics covered in this section.

Media

ACCESS ADDITIONAL ONLINE RESOURCES

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

489.

$\sqrt{36} \sqrt{36}$

490.

$\sqrt{4} \sqrt{4}$

491.

$\sqrt{64} \sqrt{64}$

492.

$\sqrt{144} \sqrt{144}$

493.

$- \sqrt{4} - \sqrt{4}$

494.

$- \sqrt{100} - \sqrt{100}$

495.

$- \sqrt{1} - \sqrt{1}$

496.

$- \sqrt{121} - \sqrt{121}$

497.

$\sqrt{−121} \sqrt{−121}$

498.

$\sqrt{−36} \sqrt{−36}$

499.

$\sqrt{−9} \sqrt{−9}$

500.

$\sqrt{−49} \sqrt{−49}$

501.

$\sqrt{9 + 16} \sqrt{9 + 16}$

502.

$\sqrt{25 + 144} \sqrt{25 + 144}$

503.

$\sqrt{9} + \sqrt{16} \sqrt{9} + \sqrt{16}$

504.

$\sqrt{25} + \sqrt{144} \sqrt{25} + \sqrt{144}$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

505.

$\sqrt{70} \sqrt{70}$

506.

$\sqrt{55} \sqrt{55}$

507.

$\sqrt{200} \sqrt{200}$

508.

$\sqrt{172} \sqrt{172}$

Approximate Square Roots with a Calculator

In the following exercises, use a calculator to approximate each square root and round to two decimal places.

509.

$\sqrt{19} \sqrt{19}$

510.

$\sqrt{21} \sqrt{21}$

511.

$\sqrt{53} \sqrt{53}$

512.

$\sqrt{47} \sqrt{47}$

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

513.

$\sqrt{y^{2}} \sqrt{y^{2}}$

514.

$\sqrt{b^{2}} \sqrt{b^{2}}$

515.

$\sqrt{49 x^{2}} \sqrt{49 x^{2}}$

516.

$\sqrt{100 y^{2}} \sqrt{100 y^{2}}$

517.

$- \sqrt{64 a^{2}} - \sqrt{64 a^{2}}$

518.

$- \sqrt{25 x^{2}} - \sqrt{25 x^{2}}$

519.

$\sqrt{144 x^{2} y^{2}} \sqrt{144 x^{2} y^{2}}$

520.

$\sqrt{196 a^{2} b^{2}} \sqrt{196 a^{2} b^{2}}$

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

521.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of $7575$ square feet. How long can a side of his garden be?

522.

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of $130130$ square feet. How long can a side of his patio be?

523.

Gravity An airplane dropped a flare from a height of $1,024 1,024$ feet above a lake. How many seconds did it take for the flare to reach the water?

524.

Gravity A hang glider dropped his cell phone from a height of $350350$ feet. How many seconds did it take for the cell phone to reach the ground?

525.

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, $4,000 4,000$ feet above the Colorado River. How many seconds did it take for the hammer to reach the river?

526.

Accident investigation The skid marks from a car involved in an accident measured $5454$ feet. What was the speed of the car before the brakes were applied?

527.

Accident investigation The skid marks from a car involved in an accident measured $216216$ feet. What was the speed of the car before the brakes were applied?

528.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was $175175$ feet. What was the speed of the vehicle before the brakes were applied?

529.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was $117117$ feet. What was the speed of the vehicle before the brakes were applied?

Everyday Math

530.

Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy $625625$ square centimeters of the designer tiles. How long can a side of the accent be?

531.

Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for $2,025 2,025$ tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?

Writing Exercises

532.

Why is there no real number equal to $\sqrt{−64} ? \sqrt{−64} ?$

533.

What is the difference between $9^{2} 9^{2}$ and $\sqrt{9} ? \sqrt{9} ?$

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?