Download Squares and Square Roots

Free Pre-Algebra
Lesson 49 ! page 1
Lesson 49
Squares and Square Roots
Back in Lesson 5 (waaaaaay back) when we were talking about the different arithmetic operations and the order of
operation, there was this text:
Ranking the Operations
The operations have a relative importance, based on how powerful they are.
Exponents and Roots* are the most powerful operations. They have the highest rank.
*The operation of raising a number to a power, also has an opposite operation, which is called
“taking the root.” You don’t have to worry about that now, we’ll cover it later in the course, but you
should know that the operations come in pairs of opposites.
Multiplication and Division are the middle rank operations. They outrank the
addition/subtraction pair, but are lower than the exponent/root pair.
Addition and Subtraction are the least powerful operations and have the lowest rank.
Here we are, later in the course. In this section we learn the square root operation and some important uses for it.
Perfect Squares
If you know the area of a square, can you figure out the side? The side is the square root of the area, because the square
is built on the side.
12 = 1
22 = 4
1=1
32 = 9
4 =2
9=3
4 2 = 16
52 = 25
16 = 4
25 = 5
If you know the side, you square it to find the area. If you know the area, you take the square root to find the side. The
notation for square root is the symbol
number.
over the number. The symbol evolved from writing an r for root before the
The square root and square are opposite operations, just like multiplication and division are opposite and addition and
subtraction are opposites.
Example: Write the related square and square root equations for 62 and 72.
(6)
2
= 36
36 = 6
Notice that the
(7 )
2
= 49
49 = 7
symbol vanishes when the operation is complete. Students sometimes write
© 2010 Cheryl Wilcox
9= 3
, but this is not
Free Pre-Algebra
Lesson 49 ! page 2
correct. The square root sign tells you to figure out what number squared is equal to 9. Once you’ve come up with 3, the
operation is completed and the action of the symbol has been accomplished, so it is not written with the answer.
To find a square root your thought process is “What number squared gives me this answer?” For example,
“I need to find the square root of 16. What number squared gives me 16 as an answer?
Ah, that must be 4. 42 = 16. The square root of 16 is 4.”
Students sometimes try dividing by 2 to find the square root, but that won’t work, because multiplying by 2 is not the same
as squaring. Squaring is multiplying a number by itself.
Example: Find the square root of each number.
36
49 = 7
=6
64 = 8
100 = 10
The Squares Between
All of the numbers 1, 4, 9, 16, 25, 36, etc are called perfect squares, because they are the result of squaring a whole
number. Squares with one of these areas will have a side that is a whole number. It’s fairly straightforward to think of the
square root for these smaller numbers. But what if the number is quite large, or is not a perfect square?
Example: What is the square root of 20?
4 2 = 16
16 = 4
?2 = 20
20 = ?
52 = 25
25 = 5
The square with area 20 cm2 does not have a whole number side. Since 20 is between the perfect squares 16 and 25, it
looks like the square root of 20 will be between 4 and 5. You could make a guess and say it’s about 4.5. To check your
guess, you could try multiplying: (4.5)(4.5) = 20.25. That’s pretty close, but a little high. On the other hand, (4.4)(4.4)
= 19.36, a little low. Looks like the square root of 20 is between 4.4 and 4.5 and probably closer to 4.5.
So far we have narrowed down the square root to the nearest tenth. Let’s try for the nearest hundredth, starting with
numbers closer to 4.5.
(4.49)(4.49) = 20.1601 (4.48)(4.48) = 20.0704 (4.47)(4.47) = 19.9809
Since 20 is between 20.0704 and 19.9809, the square root of 20 is between 4.48 and 4.47. We can continue finding decimal
places of the square root in this way. If you only have a four function calculator (or your own brain), this is a way to
approximate a square root.
Since we haven’t narrowed the square root down precisely yet, we take the two decimal places we have and round off to
one less place. Since the root is between 4.48 and 4.47, we round to the nearest tenth and use 4.5. We can’t round to either
4.48 or 4.47 because we don’t know the thousandths place.
20 ! 4.5
Note that we must use “approximately equal” because we have rounded the root. 4.5 is not the exact answer, it’s only
approximate.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 49 ! page 3
With a scientific calculator, however, you have a square root key that allows you to find as many decimal places as your
calculator displays. The square root symbol is usually above the x2 key. You access it by pressing
.
On most calculators you press
to find the square root of 20. On some
calculators you enter the number first and then the square root key. Experiment with your calculator, using some of the
perfect squares to see what works. When you evaluate the square root of 20 on your calculator, you will see:
20 = 4.472135955
If you square the number 4.472135955 on your calculator it will show 20 as the result,
meaning that the answer is 20 when rounded to as many digits your calculator
x
displays. However, if you were to laboriously multiply 4.472135955 by 4.472135955,
you’d see that it is not exactly equal to 20. You can tell if you just start the problem that
the last digit is not zero, and so the product is not exactly 20.
That means that even if we were to write all the digits displayed on the calculator, we
are still rounding, and so we have to use the “approximately equal” symbol when we
write the answer to a square root with decimals. Since even nine decimal places is not
really precise, let’s agree to round to three places unless there is some reason to go
further. Just keep in mind that the decimal answer is not precisely correct.
20 ! 4.472135955 so, rounded to three places, 20 ! 4.472
4.472135955
4.472135955
22360679775
223606797750
4024922359500
22360679775000
134164078650000
447213595500000
8944271910000000
313049516850000000
1788854382000000000
17888543820000000000
20.000000000008762025
Example: Find the square roots, and round to three decimal places if rounding is necessary.
41
2
about 6.403
0.25
exactly 0.5
© 2010 Cheryl Wilcox
1000
about 1.414
0.5
about 0.707
about 31.623
0.05
about 0.224
790321
exactly 889
5.01
about 2.238
Free Pre-Algebra
Lesson 49 ! page 4
Negative Numbers and Square Roots
So far, we’ve only considered positive numbers. The idea of squares and square roots starts with sides and areas, so we
need to extend the idea to include the negative numbers.
The first thing to recollect is that the squares of negative numbers are positive, because negative • negative = positive.
( !1) = ( !1)( !1) = 1
2
( !2) = ( !2)( !2) = 4
2
( !3)
2
= 9 , etc.
So, if we want to find a number whose square is negative, we are out of luck. There isn’t any number like that anywhere on
the number line. All the positive and negative numbers have positive squares. The square of 0 is 0. Since the square root of
a negative number is not any number on the number line, we write “not a real number” for the answer.
!9 is not a real number.
Now at this point you may be wondering why we say the
square root of 9 is 3, if (–3)(–3) = 9 also. And that’s a
very good question. Although the idea of areas and
sides was the beginning of the square root idea, we
would like to include –3 as an answer.
Every positive number has two square roots, a
positive root and a negative root.
To distinguish between the two roots, we include a
negative sign in front of the square root symbol.
9 = 3 and ! 9 = !3
That way each problem has only one answer.
If It’s Not a Real Number, What Is It?
Although there aren’t any real numbers with squares that
are negative, we could make up some numbers that have
that property. Does that seem legitimate to you? We’ve
expanded the number system from the counting numbers
to include zero, and fractions and decimals, and negatives,
so why not expand some more and make up some new
numbers?
We make up a number called i (for imaginary). The
number i is a lot like the number 1, except that i 2 = !1.
To write the square root of a negative number using
imaginary numbers, write the square root of the positive
number and follow with i:
!9 = 3i
In summary:
!25 is not a real number
25 = 5
! 25 = –5
These imaginary numbers actually have many legitimate
uses in physics. They’re not really imaginary any more.
They are usually introduced to students in Algebra II.
Example: Find the square roots if possible, and round to three decimal places if rounding is necessary.
!49
not a real number
© 2010 Cheryl Wilcox
! 2
about –1.414
! 100
–1.44
–10
not a real number
Free Pre-Algebra
Lesson 49 ! page 5
Square Root in the Order of Operations
When we use the square root with other operations, it has the same priority as exponents. Also, the root symbol can act as a
parentheses for expressions inside.
Example: Evaluate each expression.
16 + 9
16 + 9
= 4+3=7
= 25 = 5
The square root has priority over addition, so we take each
root first, then add. Note that the result is different from the
first problem.
We add the 16 and 9 first, since they are grouped by the
square root symbol. Then we take the square root of the
sum.
16 • 9
16 • 9
= 4 • 3 = 12
= 144 = 12
Here we take the square roots first. However the result is the
same as in the first problem. So it doesn’t matter with
multiplication inside square roots if we take the root first or
multiply first. This will be useful later.
The multiplication inside takes place first.
9
9
16
16
= 0.5625 = 0.75
3
= 0.75
4
=
The division is done first, then the root.
Just as with multiplication, we can evaluate the root first and
then the division, or the division first and then the root. Also
like the multiplication, this will be useful.
( )
16
2
162
( )
= 4
2
= 256 = 16
= 16
The parentheses tell us to take the square root first, then
square the result. Since square root and square are opposite
operations, it’s not really a surprise that doing one after the
other gets us back where we started.
If you square first, then take the square root, you again find
yourself back where you started. BEWARE, however,
because if you start with a negative number, you end up with
a positive, which is the absolute value of the number you
started with.
( !16)
2
!
© 2010 Cheryl Wilcox
= 256 = 16
Free Pre-Algebra
Lesson 49  page 6
Lesson 49: Squares and Square Roots Name
Worksheet
Name_________________________________________
1. Fill in the blanks in the related square and square root
problems.
2. Write the related square and square root problems for 62,
72 , 8 2 , 9 2 .
(2 )
2
=4
=2
( 3)
2
=9
9=
( )
2
= 16
16 =
( 5)
2
=
=
3. Which two perfect squares is the number between?
example:
24
24 is between 4 and 5.
a.
b.
4. Find the square roots with your calculator. Round to three
decimal places.
a.
24
b.
30
c.
75
30
75
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 49 ! page 7
5. Evaluate.
6. Evaluate.
a. ! 36
!36
b. ! 64
!64
c. ! 100
7. Evaluate.
a.
36 • 4
b.
36 • 4
c.
d.
!100
a.
100 ! 36
b.
100 ! 36
c.
25 + 144
d.
25 + 144
8. Evaluate.
a.
92
b.
( )
36
9
2
4
c.
( !9)
36
4
d.
!9
© 2010 Cheryl Wilcox
2
Free Pre-Algebra
Lesson 49 ! page 8
Lesson 49: Squares and Square Roots
Homework 49A
Name______________________________________
1. Find equivalent fractions with a common denominator.
2. Add the fractions.
7
30
7
8
+
30 45
8
45
3. A tennis ball is shot straight up at a speed of 40 ft/sec. Its
height t seconds after it is shot is given by the equation
4. A printing press can print 11,000 sheets per hour. How
long will it take to print 209,000 sheets?
h = !16t 2 + 40t + 6
What is the height of the ball after 2.6 seconds?
5. Find the perimeter of a rectangle with length that is
91/4 inches and width that is 3/4 inch more than the length.
6. Find the area of a circle with radius 44.27 cm. Round to
two decimal places.
A = !r 2
7. Find the area of a triangle with base 5/8 inch and height
that is twice as long as the base.
A=
1
bh
2
© 2010 Cheryl Wilcox
8. Find the volume of sphere with radius 0.5 cm. Round to
the nearest thousandth.
V=
4 3
!r
3
Free Pre-Algebra
Lesson 49 ! page 9
9. Rita-Marie scored 56 out of 60 possible points. What was
her percent grade?
10. Joel got 85% on the same test worth 60 points. How
many points did he get?
11. Vondell earned 138 points on an assignment, which was
92% of the possible points. How many points were possible?
12. Lillian got 80 on the first test and 76 on the second test.
What was the percent decrease in her scores?
13. Bodie got 76 on the first test and 80 on the second test.
What was the percent increase in his scores?
14. Find the square roots using your calculator. Round to
three decimal places if rounding is necessary.
a.
95
b. ! 150
c.
15. Write the related square and square root equations for
112 and 122.
361
16. Find the square roots.
a.
!121
b. ! 121
c. ! 81
17. Evaluate.
18. Evaluate.
a.
( !5)
a.
49 + 576
b.
( !5)
b.
49 + 576
c.
49 • 576
d.
49 • 576
2
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 49 ! page 10
Lesson 49: Squares and Square Roots
Homework 49A Answers
1. Find equivalent fractions with a common denominator.
2. Add the fractions.
7
7
3 21
=
• =
30 2 • 3 • 5 3 90
7
8
21 16 37
+
=
+
=
30 45 90 90 90
8
8
2 16
=
• =
45 3 • 3 • 5 2 90
3. A tennis ball is shot straight up at a speed of 40 ft/sec. Its
height t seconds after it is shot is given by the equation
4. A printing press can print 11,000 sheets per hour. How
long will it take to print 209,000 sheets?
d = rt
209,000 = 11,000t
t = 209,000 / 11,000 = 19
h = !16t 2 + 40t + 6
What is the height of the ball after 2.6 seconds?
( )
( )
= !16 ( 6.76 ) + 104 + 6 = 1.84
It will take 19 hours.
2
h = !16 2.6 + 40 2.6 + 6
The ball is 1.84 feet above ground.
5. Find the perimeter of a rectangle with length that is
91/4 inches and width that is 3/4 inch more than the length.
1 3
Width is 9 + = 10 inches.
4 4
! $
! 1
$
!
1$
77 &
P = 2 # 9 + 10& = 2 # 19 & = 2 #
# 4&
" 4
%
" 4%
" 2 %
=
1
bh
2
A = !r 2
(
A = ! 44.27
)
2
" 6277.18
The area is about 6277.18 cm2
77
1
= 38 inches
2
2
7. Find the area of a triangle with base 5/8 inch and height
that is twice as long as the base.
A=
6. Find the area of a circle with radius 44.27 cm. Round to
two decimal places.
Height = 2 •
5
8
inch =
5
inch
4
8. Find the volume of sphere with radius 0.5 cm. Round to
the nearest thousandth.
V=
( )
4 3 4
!r = ! 0.5
3
3
4
1 ! 5 $ ! 5 $ 25 2
A = # &# & =
in
2 " 8 % " 4 % 64
© 2010 Cheryl Wilcox
=
(
3
)
4
! 0.125 " 0.5235
3
Volume is about 0.524 cm3
Free Pre-Algebra
Lesson 49 ! page 11
9. Rita-Marie scored 56 out of 60 possible points. What was
her percent grade?
10. Joel got 85% on the same test worth 60 points. How
many points did he get?
56 / 60 = 0.9333…
85% of 60 points is
Her grade was 93%.
0.85 • 60 = 51
He got 51 points.
11. Vondell earned 138 points on an assignment, which was
92% of the possible points. How many points were possible?
12. Lillian got 80 on the first test and 76 on the second test.
What was the percent decrease in her scores?
92% of possible pts is 138 pts
Difference is 80 – 76 = 4 points
0.92 x = 138
4 / 80 = 0.05
x = 138/0.92 = 150
It was a 5% decrease.
There were 150 points possible.
13. Bodie got 76 on the first test and 80 on the second test.
What was the percent increase in his scores?
Difference is 80 – 76 = 4 points
4 / 76 = 0.0526315…
It was about a 5.3% increase.
14. Find the square roots using your calculator. Round to
three decimal places if rounding is necessary.
a.
b. ! 150 ! "12.247
c.
15. Write the related square and square root equations for
112 and 122.
(11)
2
= 121
(12)
121 = 11
17. Evaluate.
2
= 144
144 = 12
95 ! 9.747
361 = 19
16. Find the square roots.
a.
!121 not a real number
b. ! 121 = –11
c. ! 81 = –9
18. Evaluate.
a.
( !5)
= 25 = 5
a.
49 + 576 = 625 = 25
b.
( !5) not a real number
b.
49 + 576 = 7 + 24 = 31
c.
49 • 576 = 7 • 24 = 168
d.
49 • 576 = 282224 = 168
2
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 49 ! page 12
Lesson 49: Squares and Square Roots
Homework 49B
Name______________________________________
1. Find equivalent fractions with a common denominator.
2. Subtract the fractions.
19
30
19 11
!
30 12
11
12
3. A tennis ball is shot straight up at a speed of 50 ft/sec. Its
height t seconds after it is shot is given by the equation
4. A printing press can print 13,000 sheets per hour. How
long will it take to print 429,000 sheets?
h = !16t 2 + 50t + 3
What is the height of the ball after 3.1 seconds?
5. Find the perimeter of a rectangle with length that is
61/8 inches and width that is 5/8 inch more than the length.
6. Find the area of a circle with radius 5.01 cm. Round to two
decimal places.
A = !r 2
7. Find the area of a triangle with base 7/8 inch and height
that is twice as long as the base.
A=
1
bh
2
© 2010 Cheryl Wilcox
8. Find the volume of sphere with radius 100.5 cm. Round to
the nearest thousand.
V=
4 3
!r
3
Free Pre-Algebra
Lesson 49 ! page 13
9. Rita-Marie scored 56 out of 80 possible points. What was
her percent grade?
10. Joel got 85% on the same test worth 80 points. How
many points did he get?
11. Vondell earned 153 points on an assignment, which was
85% of the possible points. How many points were possible?
12. Lillian got 90 on the first test and 85 on the second test.
What was the percent decrease in her scores?
13. Bodie got 85 on the first test and 90 on the second test.
What was the percent increase in his scores?
14. Find the square roots using your calculator. Round to
three decimal places if rounding is necessary.
a.
45
b. ! 450
c.
15. Write the related square and square root equations for
132 and 142.
529
16. Find the square roots.
a.
!121
b. ! 121
c. ! 81
17. Evaluate.
a.
(!6)
b.
( !6)
2
© 2010 Cheryl Wilcox
18. Evaluate.
a.
64 + 225
b.
64 + 225
c.
64 • 225
d.
64 • 225