CHAPTER 7 Powers, Roots, and Scientific Notation
7.4 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
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 
.
Square of a Number
If 
, then 
is the square of .
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.
Perfect Squares
A perfect square is the square of a whole number.
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?
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 .
Square Root of a Number
A number whose square is is called a square root of .
If , then 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.
Square Root Notation
is read as “the square root of 
.
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.
EXAMPLE 1
Simplify: a) 
b) 
.
| a) | |
 |
|
Since  |
 |
| b) | |
 |
|
| Since |
 |
TRY IT 1.1
Simplify: a) 
b) 
.
Show answer
- 6
- 13
TRY IT 1.2
Simplify: a) 
b) 
.
Show answer
- 4
- 14
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, 
.
EXAMPLE 2
Simplify. a) 
b) 
| a) | |
 |
|
| The negative is in front of the radical sign. |  |
| b) | |
 |
|
| The negative is in front of the radical sign. |  |
TRY IT 2.1
Simplify: a) 
b) 
.
Show answer
- −2
- −15
TRY IT 2.2
Simplify: a) 
b) 
.
Show answer
- −9
- −8
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.
EXAMPLE 3
Simplify: a) 
b) 
.
a) There is no real number whose square is 
. Therefore, 
is not a real number.
b) The negative is in front of the radical sign, so we find the opposite of the square root of .
 |
|
| The negative is in front of the radical. | |
TRY IT 3.1
Simplify: a) 
b) 
.
Show answer
- not a real number
- −9
TRY IT 3.2
Simplify: a) 
b) .
Show answer
- −7
- not a real number
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 4
Simplify: a) 
b) 
.
| a) Use the order of operations. | |
 |
|
| Simplify each radical. |  |
| Add. |  |
| b) Use the order of operations. | |
 |
|
| Add under the radical sign. |  |
| Simplify. |  |
TRY IT 4.1
Simplify: a) 
b) 
.
Show answer
- 7
- 5
TRY IT 4.2
Simplify: a) 
b) 
.
Show answer
- 17
- 23
Notice the different answers in parts a) and b) of (Example 4). It is important to follow the order of operations correctly. In a), we took each square root first and then added them. In b), we added under the radical sign first and then found the square root.
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
EXAMPLE 5
Estimate 
between two consecutive whole numbers.
Think of the perfect squares closest to 
. Make a small table of these perfect squares and their squares roots.
 |
 |
 |
 |
TRY IT 5.1
Estimate 
between two consecutive whole numbers.
Show answer
TRY IT 5.2
Estimate 
between two consecutive whole numbers.
Show answer
The square roots of numbers that are not perfect squares are not whole numbers, they are irrational numbers. Its decimal form does not stop and does not repeat. Are irrational numbers real numbers? Yes, they are. When we put together the irrational numbers and rational numbers, we get the set of real numbers.
Let’s see how we can use calculator to find the approximate square roots of those irrational numbers.
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.
. The square root of is the example of irrational number and its approximation displays nine digits after the decimal place.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 .
EXAMPLE 6
Round 
to two decimal places using a calculator.
|
|
| Use the calculator square root key. |  |
| Round to two decimal places. |  |
 |
TRY IT 6.1
Round 
to two decimal places.
Show answer
≈ 3.32
TRY IT 6.2
Round 
to two decimal places.
Show answer
≈ 3.61
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.
EXAMPLE 7
Simplify: 
.
Think about what we would have to square to get 
. Algebraically, 
|
|
Since  |
 |
TRY IT 7.1
Simplify: 
.
Show answer
y
TRY IT 7.2
Simplify: 
.
Show answer
m
EXAMPLE 8
Simplify: 
.
|
|
 |
 |
TRY IT 8.1
Simplify: 
.
Show answer
8x
TRY IT 8.2
Simplify: 
.
Show answer
13y
EXAMPLE 9
Simplify: 
.
|
|
 |
 |
TRY IT 9.1
Simplify: 
.
Show answer
−11y
TRY IT 9.2
Simplify: 
.
Show answer
−10p
EXAMPLE 10
Simplify: 
.
|
|
 |
 |
TRY IT 10.1
Simplify: 
.
Show answer
10ab
TRY IT 10.2
Simplify: 
.
Show answer
15mn
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!
- 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.
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 the table below.
| Area (square units) | Length of side (units) |
|---|---|
|
 |
 |
 |
|
|
EXAMPLE 11
Mike and Lychelle want to make a square patio. They have enough concrete for an area of 
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 square feet and want to find the length of the side. If the area of the square is square units, the length of a side is 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. | |
Evaluate when  . |
 |
| Use your calculator. |  . |
| Round to one decimal place. |  |
| Write a sentence. | Each side of the patio should be  feet. |
TRY IT 11.1
Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 
square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?
Show answer
19.2 feet
TRY IT 11.2
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 
square centimetres. How long can a side of his mosaic be?
Show answer
52 centimetres
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 
.
|
|
| Take the square root of 64. |  |
| Simplify the fraction. | |
It would take seconds for an object dropped from a height of feet to reach the ground.
EXAMPLE 12
Christy dropped her sunglasses from a bridge 
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. | |
Evaluate when  . |
 |
| Find the square root of 400. |  |
| Simplify. | |
| Write a sentence. | It will take 5 seconds for the sunglasses to reach the river. |
TRY IT 12.1
A helicopter drops a rescue package from a height of 
feet. How many seconds does it take for the package to reach the ground?
Show answer
9 seconds
TRY IT 12.2
A window washer drops a squeegee from a platform 
feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?
Show answer
3.5 seconds
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 
.
EXAMPLE 13
After a car accident, the skid marks for one car measured 
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. | |
Evaluate when . |
 |
| Multiply. |  |
| Use your calculator. |  . |
| Round to tenths. |  |
| Write a sentence. | The speed of the car was approximately 67.5 miles per hour. |
TRY IT 13.1
An accident investigator measured the skid marks of a car and found their length was 
feet. To the nearest tenth, what was the speed of the car before the brakes were applied?
Show answer
42.7 mph
TRY IT 13.2
The skid marks of a vehicle involved in an accident were 
feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?
Show answer
54.1 mph
Key Concepts
- Square Root Notation is read ‘the square root of ’
If
, then 
, for 
. 
- 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.
Simplify Expressions with Square Roots
In the following exercises, simplify.
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8. |
9.  |
10.  |
11.  |
12.  |
13.  |
14. |
15.  |
16. |
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
17.  |
18.  |
| 19. |
20.  |
Approximate Square Roots with a Calculator
In the following exercises, use a calculator to approximate each square root and round to two decimal places.
21.  |
22.  |
23.  |
24.  |
Simplify Variable Expressions with Square Roots
In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)
| 25. |
26.  |
27.  |
28.  |
29.  |
30.  |
31.  |
32.  |
Use Square Roots in Applications
In the following exercises, solve. Round to one decimal place.
33. Landscaping Reid wants to have a square garden plot in his backyard. He has enough compost to cover an area of  square feet. How long can a side of his garden be? |
34. Landscaping Tasha wants to make a square patio in her yard. She has enough concrete to pave an area of  square feet. How long can a side of her patio be? |
35. Gravity An airplane dropped a flare from a height of  feet above a lake. How many seconds did it take for the flare to reach the water? |
36. Gravity A hang glider dropped his cell phone from a height of  feet. How many seconds did it take for the cell phone to reach the ground? |
37. Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk,  feet above the Colorado River. How many seconds did it take for the hammer to reach the river? |
38. Accident investigation The skid marks from a car involved in an accident measured  feet. What was the speed of the car before the brakes were applied? |
39. Accident investigation The skid marks from a car involved in an accident measured  feet. What was the speed of the car before the brakes were applied? |
40. 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  feet. What was the speed of the vehicle before the brakes were applied? |
41. 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  feet. What was the speed of the vehicle before the brakes were applied? |
Everyday Math
42. Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy  square centimetres of the designer tiles. How long can a side of the accent be? |
43. Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for  tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be? |
Writing Exercises
44. Why is there no real number equal to  |
45. What is the difference between  and  |
| 1. 6 | 3. 8 | 5. -2 |
| 7. -1 | 9. not a real number | 11. not a real number |
| 13. 5 | 15. 7 | 17.  |
19.  |
21. 4.36 | 23. 7.28 |
| 25. y | 27. 7x | 29. −8a |
| 31. 12xy | 33. 8.7 feet | 35. 8 seconds |
| 37. 15.8 seconds | 39. 72 mph | 41. 53.0 mph |
| 43. 45 inches | 45. Answers will vary. 92 reads: “nine squared” and means nine times itself. The expression  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. |
This chapter has been adapted from “Simplify and Use Square Roots” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.
