# Decimals

## 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  is multiplied by itself, we can write this as  which we read aloud as  For example,  is read as 
We call  the square of  because  Similarly,  is the square of  because 

### 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.
This is why we say that the square of three is nine.
The number  is called a perfect square because it is the square of a whole number.
The chart shows the squares of the counting numbers  through  You can refer to it to help you identify the perfect squares.
What happens when you square a negative number?
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  to 
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  we say  is the square of  We can also say that  is a square root of 
Notice  also, so  is also a square root of  Therefore, both  and  are square roots of  
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,  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  Notice that zero has only one square root.
The chart shows the square roots of the first  perfect square numbers.
Every positive number has two square roots and the radical sign indicates the positive one. We write  If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 

### Square Root of a Negative Number

Can we simplify  Is there a number whose square is 
None of the numbers that we have dealt with so far have a square that is  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  If we are asked to find the square root of any negative number, we say that the solution is not a real number.

### 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 . 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.
We might conclude that the square roots of numbers between  and  will be between  and  and they will not be whole numbers. Based on the pattern in the table above, we could say that  is between  and  Using inequality symbols, we write

### 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  or  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  and it is read approximately.
Suppose your calculator has a  display. Using it to find the square root of  will give  This is the approximate square root of  When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.
You will seldom use this many digits for applications in algebra. So, if you wanted to round  to two decimal places, you would write
How do we know these values are approximations and not the exact values? Look at what happens when we square them.
The squares are close, but not exactly equal, to 

### 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  where  Can you think of an expression whose square is 
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  square units, the length of a side is  units. See .

### Square Roots and Gravity

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


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

### 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  feet, then the speed of the car can be found by evaluating 

### Key Concepts

1. Square Root Notation  is read ‘the square root of ’
If , then , for .

2. Use a strategy for applications with square roots.


### Section Exercises


### Practice Makes Perfect

Simplify Expressions with Square Roots 
In the following exercises, simplify.
Estimate Square Roots 
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots with a Calculator 
In the following exercises, use a calculator to approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots 
In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)
Use Square Roots in Applications 
In the following exercises, solve. Round to one decimal place.

### Everyday Math


### Writing Exercises


### 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?

### Chapter Review Exercises


### Decimals

Name Decimals 
In the following exercises, name each decimal.
Write Decimals 
In the following exercises, write as a decimal.
Convert Decimals to Fractions or Mixed Numbers 
In the following exercises, convert each decimal to a fraction. Simplify the answer if possible.
Locate Decimals on the Number Line
Order Decimals 
In the following exercises, order each of the following pairs of numbers, using  or 
Round Decimals 
In the following exercises, round each number to the nearest: ⓐ hundredth ⓑ tenth ⓒ whole number.

### Decimal Operations

Add and Subtract Decimals 
In the following exercises, add or subtract.
Multiply Decimals 
In the following exercises, multiply.
Divide Decimals 
In the following exercises, divide.
Use Decimals in Money Applications 
In the following exercises, use the strategy for applications to solve.

### Decimals and Fractions

Convert Fractions to Decimals 
In the following exercises, convert each fraction to a decimal.
Order Decimals and Fractions 
In the following exercises, order each pair of numbers, using  or 
In the following exercises, write each set of numbers in order from least to greatest.
Simplify Expressions Using the Order of Operations 
In the following exercises, simplify
Find the Circumference and Area of Circles 
In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle.

### 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.
Solve Equations with Decimals 
In the following exercises, solve.
Translate to an Equation and Solve 
In the following exercises, translate and solve.

### Averages and Probability

Find the Mean of a Set of Numbers 
In the following exercises, find the mean of the numbers.
Find the Median of a Set of Numbers 
In the following exercises, find the median.
Find the Mode of a Set of Numbers 
In the following exercises, identify the mode of the numbers.
Use the Basic Definition of Probability 
In the following exercises, solve. (Round decimals to three places.)

### Ratios and Rate

Write a Ratio as a Fraction 
In the following exercises, write each ratio as a fraction. Simplify the answer if possible.
Write a Rate as a Fraction 
In the following exercises, write each rate as a fraction. Simplify the answer if possible.
Find Unit Rates 
In the following exercises, find the unit rate.
Find Unit Price 
In the following exercises, find the unit price.
In the following exercises, find each unit price and then determine the better buy.
Translate Phrases to Expressions with Fractions 
In the following exercises, translate the English phrase into an algebraic expression.

### Simplify and Use Square Roots

Simplify Expressions with Square Roots 
In the following exercises, simplify.
Estimate Square Roots 
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots 
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots 
In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)
Use Square Roots in Applications 
In the following exercises, solve. Round to one decimal place.

### Chapter Practice Test

In the following exercises, simplify each expression.
In the following exercises, solve.
In the following exercises, simplify.