5.7 Simplify and Use Square Roots

Simplify Expressions with Square Roots

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

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

We call $64$ the square of $8$ because $8 squared equals 64.$ Similarly, $121$ is the square of $11,$ because $11 squared equals 121.$

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.

$3 squared equals 9$

The number $9$ is called a perfect square because it is the square of a whole number.

The chart shows the squares of the counting numbers $1$ through $15 .$ You can refer to it to help you identify the perfect squares.

A table with two columns is shown. The first column is labeled “Number” and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled “Square” and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

What happens when you square a negative number?

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

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 $−1$ to $−15.$

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled “Square” and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

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 $10 squared equals 100,$ we say $100$ is the square of $10 .$ We can also say that $10$ is a square root of $100 .$

Notice $open parenthesis −10 close parenthesis squared equals 100$ also, so $−10$ is also a square root of $100 .$ Therefore, both $10$ and $−10$ are square roots of $100 .$

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, $\sqrt{},$ stands for the positive square root. The positive square root is also called the principal square root.

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

The chart shows the square roots of the first $15$ perfect square numbers.

A table is shown with 2 columns. The first column contains the values: square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second column contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.

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

Square Root of a Negative Number

Can we simplify $the square root of −25 ?$ Is there a number whose square is $−25 ?$

$open parenthesis close parenthesis squared equals −25 ?$

None of the numbers that we have dealt with so far have a square that is $−25.$ 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 $the square root of −25.$ If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Example 3

Simplify: ⓐ $the square root of −169$ ⓑ $- \sqrt{121} .$

ⓐ There is no real number whose square is $−169.$ Therefore, $the square root of −169$ is not a real number.

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

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

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.

Notice the different answers in parts ⓐ and ⓑ of Example 4. 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 $4$ and $9$ will be between $2$ and $3,$ and they will not be whole numbers. Based on the pattern in the table above, we could say that $the square root of 5$ is between $2$ and $3 .$ Using inequality symbols, we write

$2 less than the square root of 5 less than 3$

Example 5

Estimate $the square root of 60$ between two consecutive whole numbers.

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

$Locate 60 between two consecutive perfect squares.$	$49 less than 60 less than 64$
$the square root of 60 is between their square roots.$	$7 less than the square root of 60 less than 8$

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 $\sqrt{}$ or $\sqrt{x}$ key on your calculator. You will need 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 $\approx$ and it is read approximately.

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

$\sqrt{5} \approx 2.236067978$

You will seldom use this many digits for applications in algebra. So, if you wanted to round $the square root of 5$ to two decimal places, you would write

$\sqrt{5} \approx 2.24$

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

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

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

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 $the square root of 9 x squared,$ where $x greater than or equal to 0.$ Can you think of an expression whose square is $9 x squared ?$

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

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.

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!

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 $A$ square units, the length of a side is $\sqrt{A}$ units. See Table 5.8.

**Table 5.8**
Area (square units)	Length of side (units)
$9$	$\sqrt{9} = 3$
$144$	$\sqrt{144} = 12$
$A$	$\sqrt{A}$

Example 11

Mike and Lychelle want to make a square patio. They have enough concrete for an area of $200$ square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

We know the area of the square is $200$ square feet and want to find the length of the side. If the area of the square is $A$ square units, the length of a side is $\sqrt{A}$ 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.	$\sqrt{A}$
Evaluate $\sqrt{A}$ when $A equals 200$ .	$the square root of 200$
Use your calculator.	$14.142135...$
Round to one decimal place.	$14.1 feet$
Write a sentence.	Each side of the patio should be $14.1$ feet.

Square Roots and Gravity

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

	$\frac{\sqrt{64}}{4}$
Take the square root of 64.	$\frac{8}{4}$
Simplify the fraction.	$2$

It would take $2$ seconds for an object dropped from a height of $64$ feet to reach the ground.

Example 12

Christy dropped her sunglasses from a bridge $400$ feet above a river. How many seconds does it take for the sunglasses to reach the river?

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.	$\frac{\sqrt{h}}{4}$
Evaluate $\frac{\sqrt{h}}{4}$ when $h equals 400$ .	$\frac{\sqrt{400}}{4}$
Find the square root of 400.	$\frac{20}{4}$
Simplify.	$5$
Write a sentence.	It will take 5 seconds for the sunglasses to reach the river.

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 $d$ feet, then the speed of the car can be found by evaluating $the square root of 24 d.$

Example 13

After a car accident, the skid marks for one car measured $190$ feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?

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.	$the square root of 24 d$
Evaluate $the square root of 24 d$ when $d equals 190.$	$\sqrt{24 \cdot 190}$
Multiply.	$the square root of 4,560$
Use your calculator.	$67.527772...$
Round to tenths.	$67.5$
Write a sentence.	The speed of the car was approximately 67.5 miles per hour.

Key Concepts

Square Root Notation $\sqrt{m}$ is read ‘the square root of $m$ ’
If $m = n^{2}$ , then $\sqrt{m} = n$ , for $n greater than or equal to 0$ .
Use a strategy for applications with square roots.
- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Write a complete sentence that answers the question.

Section Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

$the square root of 36$

$the square root of 4$

$the square root of 64$

$the square root of 144$

$- \sqrt{4}$

−2

$- \sqrt{100}$

$- \sqrt{1}$

−1

$- \sqrt{121}$

$the square root of −121$

not a real number

$the square root of −36$

$the square root of −9$

not a real number

$the square root of −49$

$\sqrt{9 + 16}$

$\sqrt{25 + 144}$

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

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

Estimate Square Roots

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

$the square root of 70$

$8 less than the square root of 70 less than 9$

$the square root of 55$

$the square root of 200$

$14 less than the square root of 200 less than 15$

$the square root of 172$

Approximate Square Roots with a Calculator

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

$the square root of 19$

4.36

$the square root of 21$

$the square root of 53$

7.28

$the square root of 47$

Simplify Variable Expressions with Square Roots

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

$\sqrt{y^{2}}$

$\sqrt{b^{2}}$

$the square root of 49 x squared$

$the square root of 100 y squared$

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

−8a

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

$the square root of 144 x squared y squared$

12xy

$the square root of 196 a squared b squared$

Use Square Roots in Applications

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

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of $75$ square feet. How long can a side of his garden be?

8.7 feet

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of $130$ square feet. How long can a side of his patio be?

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

8 seconds

Gravity A hang glider dropped his cell phone from a height of $350$ feet. How many seconds did it take for the cell phone to reach the ground?

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

15.8 seconds

Accident investigation The skid marks from a car involved in an accident measured $54$ feet. What was the speed of the car before the brakes were applied?

Accident investigation The skid marks from a car involved in an accident measured $216$ feet. What was the speed of the car before the brakes were applied?

72 mph

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 $175$ feet. What was the speed of the vehicle before the brakes were applied?

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 $117$ feet. What was the speed of the vehicle before the brakes were applied?

53.0 mph

Everyday Math

Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy $625$ square centimeters of the designer tiles. How long can a side of the accent be?

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

45 inches

Writing Exercises

Why is there no real number equal to $the square root of −64 ?$

What is the difference between $9 squared$ and $the square root of 9 ?$

Answers will vary. 9² reads: “nine squared” and means nine times itself. The expression $the square root of 9$ reads: “the square root of nine” which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.

Self Check

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

A self-assessment table for square root skills, with columns for 'Confidently,' 'With some help,' and 'No-I don't get it!' for topics like simplifying, estimating, approximating, and applications. — **Figure 5.16**

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

Chapter Review Exercises

Decimals

Name Decimals

In the following exercises, name each decimal.

$0.8$

$0.375$

three hundred seventy-five thousandths

$0.007$

$5.24$

five and twenty-four hundredths

$−12.5632$

$−4.09$

negative four and nine hundredths

Write Decimals

In the following exercises, write as a decimal.

three tenths

nine hundredths

0.09

twenty-seven hundredths

ten and thirty-five thousandths

10.035

negative twenty and three tenths

negative five hundredths

−0.05

Convert Decimals to Fractions or Mixed Numbers

In the following exercises, convert each decimal to a fraction. Simplify the answer if possible.

$0.43$

$0.825$

$\frac{33}{40}$

$9.7$

$3.64$

$3 \frac{16}{25}$

Locate Decimals on the Number Line

ⓐ $0.6$

ⓑ $−0.9$

ⓓ $−1.3$

Order Decimals

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

$0.6 ___ 0.8$

$0.2 ___ 0.15$

$0.803 ____ 0.83$

$−0.56 ____ −0.562$

Round Decimals

In the following exercises, round each number to the nearest: ⓐ hundredth ⓑ tenth ⓒ whole number.

$12.529$

ⓐ 12.53
ⓑ 12.5
ⓒ 13

$4.8447$

$5.897$

ⓐ 5.90
ⓑ 5.9
ⓒ 6

Decimal Operations

Add and Subtract Decimals

In the following exercises, add or subtract.

$5.75 plus 8.46$

$32.89 minus 8.22$

24.67

$24 minus 19.31$

$10.2 plus 14.631$

24.831

$−6.4 plus open parenthesis −2.9 close parenthesis$

$1.83 minus 4.2$

−2.37

Multiply Decimals

In the following exercises, multiply.

$open parenthesis 0.3 close parenthesis open parenthesis 0.7 close parenthesis$

$open parenthesis −6.4 close parenthesis open parenthesis 0.25 close parenthesis$

−1.6

$open parenthesis −3.35 close parenthesis open parenthesis −12.7 close parenthesis$

$open parenthesis 15.4 close parenthesis open parenthesis 1000 close parenthesis$

15,400

Divide Decimals

In the following exercises, divide.

$0.48 divided by 6$

$4.32 divided by 24$

0.18

$$6.29 divided by 12$

$open parenthesis −0.8 close parenthesis divided by open parenthesis −0.2 close parenthesis$

$1.65 divided by 0.15$

$9 divided by 0.045$

200

Use Decimals in Money Applications

In the following exercises, use the strategy for applications to solve.

Miranda got $$40$ from her ATM. She spent $$9.32$ on lunch and $$16.99$ on a book. How much money did she have left? Round to the nearest cent if necessary.

Jessie put $8$ gallons of gas in her car. One gallon of gas costs $$3.528.$ How much did Jessie owe for all the gas?

$28.22

A pack of $16$ water bottles cost $$6.72.$ How much did each bottle cost?

Alice bought a roll of paper towels that cost $$2.49.$ She had a coupon for $$0.35$ off, and the store doubled the coupon. How much did Alice pay for the paper towels?

$1.79

Decimals and Fractions

Convert Fractions to Decimals

In the following exercises, convert each fraction to a decimal.

$\frac{3}{5}$

$\frac{7}{8}$

0.875

$- \frac{19}{20}$

$- \frac{21}{4}$

−5.25

$\frac{1}{3}$

$\frac{6}{11}$

$0. 54 —$

Order Decimals and Fractions

In the following exercises, order each pair of numbers, using $<$ or $>.$

$\frac{1}{2}___0.2$

$\frac{3}{5}___0 .$

$- \frac{7}{8}___−0.84$

$- \frac{5}{12}___−0.42$

$0.625___\frac{13}{20}$

$0.33___\frac{5}{16}$

In the following exercises, write each set of numbers in order from least to greatest.

$\frac{2}{3}, \frac{17}{20}, 0.65$

$\frac{7}{9}, 0.75, \frac{11}{15}$

$\frac{11}{15}, 0.75, \frac{7}{9}$

Simplify Expressions Using the Order of Operations

In the following exercises, simplify

$4 open parenthesis 10.3 minus 5.8 close parenthesis$

$\frac{3}{4} (15.44 - 7.4)$

6.03

$30 \div (0.45 + 0.15)$

$1.6 + \frac{3}{8}$

1.975

$52 open parenthesis 0.5 close parenthesis plus open parenthesis 0.4 close parenthesis squared$

$- \frac{2}{5} \cdot \frac{9}{10} + 0.14$

−0.22

Find the Circumference and Area of Circles

In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle.

$radius equals 6 in.$

$radius equals 3.5 ft.$

ⓐ 21.98 ft.
ⓑ 38.465 sq.ft.

$radius = \frac{7}{33} m$

$diameter equals 11 cm$

ⓐ 34.54 cm
ⓑ 94.985 sq.cm

Solve Equations with Decimals

Determine Whether a Decimal is a Solution of an Equation

In the following exercises, determine whether the each number is a solution of the given equation.

$x - 0.4 = 2.1$
ⓐ $x equals 1.7$ ⓑ $x equals 2.5$

$y + 3.2 = −1.5$
ⓐ $y equals 1.7$ ⓑ $y equals −4.7$

ⓐ no
ⓑ yes

$\frac{u}{2.5} = −12.5$
ⓐ $u equals −5$ ⓑ $u equals −31.25$

$0.45 v equals −40.5$
ⓐ $v equals −18.225$ ⓑ $v equals −90$

ⓐ no
ⓑ yes

Solve Equations with Decimals

In the following exercises, solve.

$m + 3.8 = 7.5$

$h + 5.91 = 2.4$

h = −3.51

$a + 2.26 = −1.1$

$p - 4.3 = −1.65$

p = 2.65

$x - 0.24 = −8.6$

$j - 7.42 = −3.7$

j = 3.72

$0.6 p equals 13.2$

$−8.6 x equals 34.4$

x = −4

$−22.32 equals −2.4 z$

$\frac{a}{0.3} = −24$

a = −7.2

$\frac{p}{−7} = −4.2$

$\frac{s}{−2.5} = −10$

s = 25

Translate to an Equation and Solve

In the following exercises, translate and solve.

The difference of $n$ and $15.2$ is $4.4 .$

The product of $−5.9$ and $x$ is $−3.54.$

−5.9x = −3.54; x = 0.6

The quotient of $y$ and $−1.8$ is $−9.$

The sum of $m$ and $open parenthesis −4.03 close parenthesis$ is $6.8 .$

m + (−4.03) = 6.8; m = 10.83

Averages and Probability

Find the Mean of a Set of Numbers

In the following exercises, find the mean of the numbers.

$2, 4, 1, 0, 1, and 1$

$$270$ , $$310.50$ , $$243.75$ , and $$252.15$

$269.10

Each workday last week, Yoshie kept track of the number of minutes she had to wait for the bus. She waited $3, 0, 8, 1, and 8$ minutes. Find the mean.

In the last three months, Raul’s water bills were $$31.45, $48.76, and $42.60.$ Find the mean.

$40.94

Find the Median of a Set of Numbers

In the following exercises, find the median.

$41$ , $45$ , $32$ , $60$ , $58$

$25$ , $23$ , $24$ , $26$ , $29$ , $19$ , $18$ , $32$

24.5

The ages of the eight men in Jerry’s model train club are $52, 63, 45, 51, 55, 75, 60, and 59.$ Find the median age.

The number of clients at Miranda’s beauty salon each weekday last week were $18, 7, 12, 16, and 20.$ Find the median number of clients.

16 clients

Find the Mode of a Set of Numbers

In the following exercises, identify the mode of the numbers.

$6$ , $4$ , $4, 5$ , $6, 6$ , $4$ , $4$ , $4$ , $3$ , $5$

The number of siblings of a group of students: $2$ , $0$ , $3$ , $2$ , $4$ , $1$ , $6$ , $5$ , $4$ , $1$ , $2$ , $3$

Use the Basic Definition of Probability

In the following exercises, solve. (Round decimals to three places.)

The Sustainability Club sells $200$ tickets to a raffle, and Albert buys one ticket. One ticket will be selected at random to win the grand prize. Find the probability Albert will win the grand prize. Express your answer as a fraction and as a decimal.

Luc has to read $3$ novels and $12$ short stories for his literature class. The professor will choose one reading at random for the final exam. Find the probability that the professor will choose a novel for the final exam. Express your answer as a fraction and as a decimal.

$\frac{1}{5}; 0.2$

Ratios and Rate

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction. Simplify the answer if possible.

$28$ to $40$

$56$ to $32$

$\frac{7}{4}$

$3.5$ to $0.5$

$1.2$ to $1.8$

$\frac{2}{3}$

$1 \frac{3}{4} to 1 \frac{5}{8}$

$2 \frac{1}{3} to 5 \frac{1}{4}$

$\frac{4}{9}$

$64$ ounces to $30$ ounces

$28$ inches to $3$ feet

$\frac{7}{9}$

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction. Simplify the answer if possible.

$180$ calories per $8$ ounces

$90$ pounds per $7.5$ square inches

$\frac{12 pounds}{1 square inch}$

$126$ miles in $4$ hours

$$612.50$ for $35$ hours

$\frac{$35}{2 hours}$

Find Unit Rates

In the following exercises, find the unit rate.

$180$ calories per $8$ ounces

$90$ pounds per $7.5$ square inches

12 pounds/sq.in.

$126$ miles in $4$ hours

$$612.50$ for $35$ hours

$17.50/hour

Find Unit Price

In the following exercises, find the unit price.

t-shirts: $3$ for $$8.97$

Highlighters: $6$ for $$2.52$

$0.42

An office supply store sells a box of pens for $$11.$ The box contains $12$ pens. How much does each pen cost?

Anna bought a pack of $8$ kitchen towels for $$13.20.$ How much did each towel cost? Round to the nearest cent if necessary.

$1.65

In the following exercises, find each unit price and then determine the better buy.

Shampoo: $12$ ounces for $$4.29$ or $22$ ounces for $$7.29 ?$

Vitamins: $60$ tablets for $$6.49$ or $100$ for $$11.99 ?$

$0.11, $0.12; 60 tablets for $6.49

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

$535$ miles per $h hours$

$a$ adults to $45$ children

$\frac{a adults}{45 children}$

the ratio of $4 y$ and the difference of $x$ and $10$

the ratio of $19$ and the sum of $3$ and $n$

$\frac{19}{3 + n}$

Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

$the square root of 64$

$the square root of 144$

$- \sqrt{25}$

$- \sqrt{81}$

−9

$the square root of −9$

$the square root of −36$

not a real number

$\sqrt{64} + \sqrt{225}$

$\sqrt{64 + 225}$

Estimate Square Roots

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

$the square root of 28$

$the square root of 155$

$12 less than the square root of 155 less than 13$

Approximate Square Roots

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

$the square root of 15$

$the square root of 57$

7.55

Simplify Variable Expressions with Square Roots

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

$\sqrt{q^{2}}$

$the square root of 64 b squared$

$- \sqrt{121 a^{2}}$

$the square root of 225 m squared n squared$

15mn

$- \sqrt{100 q^{2}}$

$the square root of 49 y squared$

$the square root of 4 a squared b squared$

$the square root of 121 c squared d squared$

11cd

Use Square Roots in Applications

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

Art Diego has $225$ square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be?

Landscaping Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of $30$ square feet. How long can a side of the flower garden be?

5.5 feet

Gravity A hiker dropped a granola bar from a lookout spot $576$ feet above a valley. How long did it take the granola bar to reach the valley floor?

Accident investigation The skid marks of a car involved in an accident were $216$ feet. How fast had the car been going before applying the brakes?

72 mph

Chapter Practice Test

Write six and thirty-four thousandths as a decimal.

Write $1.73$ as a fraction.

$1 \frac{73}{100}$

Write $\frac{5}{8}$ as a decimal.

ⓐ 16.7
ⓑ 16.75
ⓒ 17

Write the numbers $\frac{4}{5}, −0.1, 0.804, \frac{2}{9}, −7.4, 0.21$ in order from smallest to largest.

In the following exercises, simplify each expression.

$15.4 plus 3.02$

18.42

$20 minus 5.71$

$open parenthesis 0.64 close parenthesis open parenthesis 0.3 close parenthesis$

0.192

$open parenthesis −4.2 close parenthesis open parenthesis 100 close parenthesis$

$0.96 divided by open parenthesis −12 close parenthesis$

−0.08

$−5 divided by 0.025$

$−0.6 divided by open parenthesis −0.3 close parenthesis$

$open parenthesis 0.7 close parenthesis squared$

$24 \div (0.1 + 0.02)$

200

$4 open parenthesis 10.3 minus 5.8 close parenthesis$

$1.6 + \frac{3}{8}$

1.975

$\frac{2}{3} (14.65 - 4.6)$

In the following exercises, solve.

$m + 3.7 = 2.5$

−1.2

$\frac{h}{0.5} = 4.38$

$−6.5 y equals −57.2$

8.8

$1.94 = a - 2.6$

Three friends went out to dinner and agreed to split the bill evenly. The bill was $$79.35.$ How much should each person pay?

$26.45

A circle has radius $12 .$ Find the ⓐ circumference and ⓑ area. $[ Use 3.14 for π.]$

The ages, in months, of $10$ children in a preschool class are:
$55$ , $55$ , $50$ , $51$ , $52$ , $50$ , $53$ , $51$ , $55$ , $49$
Find the ⓐ mean ⓑ median ⓒ mode

ⓐ 52.1
ⓑ 51.5
ⓒ 55

Of the $16$ nurses in Doreen’s department, $12$ are women and $4$ are men. One of the nurses will be assigned at random to work an extra shift next week. ⓐ Find the probability a woman nurse will be assigned the extra shift. ⓑ Convert the fraction to a decimal.

Find each unit price and then the better buy.
Laundry detergent: $64$ ounces for $$10.99$ or $48$ ounces for $$8.49$

The unit prices are $0.172 per ounce for 64 ounces, and $0.177 per ounce for 48 ounces; 64 ounces is the better buy.

In the following exercises, simplify.

$\sqrt{36 + 64}$

$the square root of 144 n squared$

12n

Estimate $the square root of 54$ to between two whole numbers.

Yanet wants a square patio in her backyard. She has $225$ square feet of tile. How long can a side of the patio be?

15 feet

ⓑ
	$the square root of 121$
Since $11 squared equals 121$	$11$

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

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

ⓐ Use the order of operations.
	$\sqrt{25} + \sqrt{144}$
Simplify each radical.	$5 plus 12$
Add.	$17$

ⓑ Use the order of operations.
	$\sqrt{25 + 144}$
Add under the radical sign.	$the square root of 169$
Simplify.	$13$

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