New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
This material is made freely available at www.njctl.org and
is intended for the non-commercial use of students and
teachers. These materials may not be used for any
commercial purpose without the written permission of the
owners. NJCTL maintains its website for the convenience
of teachers who wish to make their work available to other
teachers, participate in a virtual professional learning
community, and/or provide access to course materials to
parents, students and others.
Click to go to website:
www.njctl.org
8th Grade Math
Numerical Roots &
Radicals
2012-12-03
www.njctl.org
Setting the PowerPoint View
Use Normal View for the Interactive Elements
To use the interactive elements in this presentation, do not select the
Slide Show view. Instead, select Normal view and follow these steps to
set the view as large as possible:
• On the View menu, select Normal.
• Close the Slides tab on the left.
• In the upper right corner next to the Help button, click the ^ to minimize
the ribbon at the top of the screen.
• On the View menu, confirm that Ruler is deselected.
• On the View tab, click Fit to Window.
• On the View tab, click Slide Master | Page Setup. Select On-screen
Show (4:3) under Slide sized for and click Close Master View.
• On the Slide Show menu, confirm that Resolution is set to 1024x768.
Use Slide Show View to Administer Assessment Items
To administer the numbered assessment items in this presentation, use
the Slide Show view. (See Slide 8 for an example.)
Numerical Roots and Radicals
Squares, Square Roots & Perfect Squares
Squares of Numbers Greater than 20
Simplifying Perfect Square Radical Expressions
Approximating Square Roots
Rational & Irrational Numbers
Radical Expressions Containing Variables
Simplifying Non-Perfect Square Radicands
Simplifying Roots of Variables
Properties of Exponents
Solving Equations with Perfect Square & Cube Roots
Common Core Standards: 8.NS.1-2; 8.EE.1-2
Click on topic
to go to that section.
Squares, Square Roots and
Perfect Squares
Return to Table of
Contents
Area of a Square
The area of a figure is the number of square units needed to
cover the figure.
The area of the square below is 16 square units because 16 square
units are needed to COVER the figure...
Area of a Square
The area (A) of a square can be found by squaring its side length, as
shown below:
2
A=s
Click 2to see if the
Aanswer
= 4 =found
4 4with
formula is
=the16Area
sq.units
correct!
The area (A) of a square is labeled as
2
square units, or units , because you
cover the figure with squares...
4 units
1
What is the area of a square with sides of 5
inches?
A
16 in2
B
20 in2
C
25 in2
D
30 in2
2
What is the area of a square with sides of 6
inches?
A
16 in2
B
20 in2
C
24 in2
D
36 in2
3
2
If a square has an area of 9 ft , what is the
length of a side?
A
2 ft
B
2.25 ft
C
3 ft
D
4.5 ft
4
What is the area of a square with a side length
of 16 in?
5
What is the side length of a square with an
area of 196 square feet?
When you square a number you multiply it by itself.
2
5 = 5  5 = 25 so the square of 5 is 25.
You can indicate squaring a number with an exponent of 2, by asking
for the square of a number, or by asking for a number squared.
What is the square of seven?
49
What is nine squared?
81
Make a list of the numbers 1-15 and then square each of them.
Your paper should be set up as follows:
Number
1
2
3
(and so on)
Square
1
4
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
The numbers in the right column are
squares of the numbers in the left column.
If you want to "undo" squaring a number,
you must take the square root of the
number.
So, the numbers in the left column are the
square roots of the numbers in the right
column.
Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
The square root of a number is found by
undoing the squaring. The symbol for
square root is called a radical sign and it
looks like this:
Using our list, to find the square root of a
number, you find the number in the right
hand column and look to the left.
So, the
81 = 9
What is
169?
Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
When the square root of a number is a
whole number, the number is called a
perfect square.
Since all of the numbers in the right
hand column have whole numbers for
their square roots, this is a list of the
first 15 perfect squares.
Find the following.
You may refer to your chart if you need to.
6
What is
1 ?
7
What is
81?
8
What is the square of 15 ?
9
What is
256?
10
2
What is 13 ?
11
What is
196?
12
What is the square of 18?
13
What is 11 squared?
14
What is 20 squared?
Squares of Numbers
Greater than 20
Return to Table of
Contents
Think about this...
What about larger numbers?
How do you find
?
It helps to know the squares of larger numbers such as the multiples
of tens.
2
10 = 100
2
202 = 400
302 = 900
402 = 1600
502 = 2500
602 = 3600
702 = 4900
802 = 6400
90 2= 8100
100 = 10000
What pattern do you notice?
For larger numbers, determine between which two multiples of ten
the number lies.
2
102 = 100
202 = 400
302 = 900
402 = 1600
50 = 2500
2
602 = 3600
702 = 4900
802 = 6400
90 2= 8100
100 = 10000
2
12 = 1
22 = 4
32 = 9
42 = 16
5 = 25
2
62 = 36
72 = 49
82 = 64
9 2= 81
10 = 100
Next, look at the ones digit to determine the ones digit of your
square root.
Examples:
2809
Lies between 2500 & 3600 (50 and 60)
Ends in nine so square root ends in 3 or 7
Try 53 then 57
2
53 = 2809
7744
Lies between 6400 and 8100 (80 and 90)
Ends in 4 so square root ends in 2 or 8
Try 82 then 88
2
82 = 6724 NO!
2
88 = 7744
15
Find.
16
Find.
17
Find.
18
Find.
19
Find.
20
Find.
21
Find.
22
Find.
23
Find.
Simplifying Perfect Square
Radical Expressions
Return to Table of
Contents
Can you recall the perfect squares from 1 to 400?
2
1 =
2
2 =
2
2
8 =
2
9 =
2
2
15 =
2
16 =
2
3 =
10 =
17 =
2
2
2
4 =
2
5 =
2
6 =
2
7 =
11 =
2
12 =
2
13 =
2=
14
18 =
2
19 =
2
20 =
Square Root Of A Number
2
Recall: If b = a, then b is a square root of a.
2
Example: If 4 = 16, then 4 is a square root of 16
What is a square root of 25? 64? 100?
5
8
10
Square Root Of A Number
Square roots are written with a radical symbol
Positive square root:
Negative square root:-
=4
= -4
Positive & negative square roots:
= 4
Negative numbers have no real square roots
no real roots because there is no real number that,
when squared, would equal -16.
Is there a difference between
&
Which expression has no real roots?
Evaluate the expressions:
?
Evaluate the expression
is not real
24
25
?
26
= ?
27
28
= ?
29
A
3
B
-3
C
No real roots
30
The expression equal to
positive integer when b is
is equivalent to a
A -10
B 64
C
16
D
4
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Square Roots of Fractions
a
b
16
49
=
=
b
=
4
7
0
Try These
31
A
C
B
D
no real solution
32
A
C
B
D
no real solution
33
A
C
B
D
no real solution
34
A
C
B
D
no real solution
35
A
C
B
D
no real solution
Square Roots of Decimals
Recall:
To find the square root of a decimal, convert the decimal to a
fraction first. Follow your steps for square roots of fractions.
= .2
= .05
= .3
36
Evaluate
A
C
B
D
no real solution
37
Evaluate
A
.06
B
.6
C
6
D
No Real Solution
38
Evaluate
A
.11
B
11
C
1.1
D
No Real Solution
39
Evaluate
A
C
.8
B
.08
D
No Real Solution
40
Evaluate
A
B
C
D
No Real Solution
Approximating
Square Roots
Return to Table of
Contents
All of the examples so far have
been from perfect squares.
What does it mean to be a perfect square?
The square of an integer is a perfect square.
A perfect square has a whole number square root.
You know how to find the square root of a perfect square.
What happens if the number is not a perfect square?
Does it have a square root?
What would the square root look like?
Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Think about the square root of 50.
Where would it be on this chart?
What can you say about the square root of
50?
50 is between the perfect squares 49 and
64 but closer to 49.
So the square root of 50 is between 7 and 8
but closer to 7.
Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
When estimating square roots of numbers,
you need to determine:
Between which two perfect squares it lies
(and therefore which 2 square roots).
Which perfect square it is closer to (and
therefore which square root).
Example: 110
Lies between 100 & 121, closer to 100.
So 110 is between 10 & 11, closer to 10.
Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Estimate the following:
30
200
215
Approximating a Square Root
Approximate
6
to the nearest integer
<
<
<
<
Identify perfect squares closest to 38
7
Take square root
Answer: Because 38 is closer to 36 than to 49,
7. So, to the nearest integer,
=6
is closer to 6 than to
Approximate
to the nearest integer
<
<
Identify perfect squares closest to 70
<
<
Take square root
Identify nearest integer
Another way to think about it is to use a number line.
√8
2
2.2
2.1
2.4
2.3
2.6
2.5
2.8
2.7
3.0
2.9
Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8
Example:
Approximate
10
10.2 10.4 10.6 10.8
11.0
10.1
10.3 10.5 10.7
10.9
41
The square root of 40 falls between which two
perfect squares?
A
9 and 16
B
25 and 36
C
36 and 49
D
49 and 64
42
Which whole number is
40 closest to?
<
<
Identify perfect squares closest to 40
<
<
Take square root
Identify nearest integer
43
The square root of 110 falls between which two
perfect squares?
A
36 and 49
B
49 and 64
C
64 and 84
D
100 and 121
44
Estimate to the nearest whole number.
110
45
Estimate to the nearest whole number.
219
46
Estimate to the nearest whole number.
90
47
What is the square root of 400?
48
Approximate
to the nearest integer.
49
Approximate
96 to the nearest integer.
50
Approximate 167 to the nearest integer.
51
Approximate 140 to the nearest integer.
52
Approximate 40 to the nearest integer.
53
The expression
A
3 and 9
B
8 and 9
C
9 and 10
D
46 and 47
is a number between
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Rational & Irrational
Numbers
Return to Table of
Contents
Rational & Irrational Numbers
is rational because the radicand (number under the radical) is a
perfect square
If a radicand is not a perfect square, the root is said to be irrational.
Ex:
Sort the following numbers.
0
24
25
300
Rational
32
36
625
40
1225
52
64
100
1681
3600
Irrational
200
225
54
Rational or Irrational?
A
Rational
B
Irrational
55
Rational or Irrational?
A
Rational
B
Irrational
56
Rational or Irrational?
A
Rational
B
Irrational
57
Rational or Irrational?
A
Rational
B
Irrational
58
Rational or Irrational?
A
Rational
B
Irrational
59
Which is a rational number?
A
B
p
C
D
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
60
Given the statement: “If x is a rational number, then
is irrational.” Which value of x makes the statement
false?
A
B
2
C
3
D
4
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Radical Expressions
Containing Variables
Return to Table of
Contents
Square Roots of Variables
To take the square root of a variable rewrite its exponent as the square
of a power.
12
=
(x12)2 = x
=
(a )
8 2
8
=a
Square Roots of Variables
If the square root of a variable raised to an even power has a variable
raised to an odd power for an answer, the answer must have absolute
value signs. This ensures that the answer will be positive.
By Definition...
Examples
Try These.
5
= |x|
13
= |x|
How many of these expressions will need an absolute value
sign when simplified?
yes
yes
no
no
yes
yes
61
Simplify
A
B
C
D
62
Simplify
A
B
C
D
63
Simplify
A
B
C
D
64
Simplify
A
B
C
D
65
A
C
B
D
no real solution
Simplifying Non-Perfect Square
Radicands
Return to Table of
Contents
What happens when the radicand is not a perfect square?
Rewrite the radicand as a product of its largest perfect square factor.
Simplify the square root of the perfect square.
When simplified form still contains a radical, it is said to be
irrational.
Try These.
Identifying the largest perfect square factor when simplifying radicals
will result in the least amount of work.
Ex:
Not simplified! Keep going!
Finding the largest perfect square factor results in less work:
Note that the answers are the same for both solution processes
66
Simplify
A
B
C
D
already in simplified form
67
Simplify
A
B
C
D
already in simplified form
68
Simplify
A
B
C
D
already in simplified form
69
Simplify
A
B
C
D
already in simplified form
70
Simplify
A
B
C
D
already in simplified form
71
Simplify
A
B
C
D
already in simplified form
72
Which of the following does not have an irrational
simplified form?
A
B
C
D
Note - If a radical begins with a coefficient before the radicand is simplified,
any perfect square that is simplified will be multiplied by the existing
coefficient. (multiply the outside)
Express
in simplest radical form.
73
Simplify
A
B
C
D
74
Simplify
A
B
C
D
75
Simplify
A
B
C
D
76
Simplify
A
B
C
D
77
Simplify
A
B
C
D
78
When
is
is written in simplest radical form, the result
. What is the value of k?
A
20
B
10
C
7
D
4
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
79
When
is expressed in simplest
form, what is the value of a?
A
6
B
2
C
3
D
8
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Simplifying Roots of Variables
Return to Table of
Contents
Simplifying Roots of Variables
Remember, when working with square roots, an absolute value sign
is needed if:
• the power of the given variable is even
and
• the answer contains a variable raised to an odd power outside
the radical
Examples of when absolute values are needed:
Simplifying Roots of Variables
Divide the exponent by 2. The number of times that 2 goes into the
exponent becomes the power on the outside of the radical and the
remainder is the power of the radicand.
Note:
Absolute value signs are not needed because the radicand had an
odd power to start.
Example
Simplify
Only the y has an odd power on the
outside of the radical.
The x had an odd power under the radical so
no absolute value signs needed.
The m's starting power was odd, so it does not
require absolute value signs.
80
Simplify
A
B
C
D
81
Simplify
A
B
C
D
82
Simplify
A
B
C
D
83
Simplify
A
B
C
D
Properties of Exponents
Return to Table of
Contents
Rules of Exponents
Materials
Exponential Table Questions.pdf
Exponential Table.pdf
Exponential Test Review.pdf
There are handouts that can be used along with this section. They are
located under the heading labs on the Exponential page of PMI Algebra.
Documents are linked. Click the name above of the document.
The Exponential Table
x
1
2
3
4
5
6
7
8
1X
2X
3X
4X
5X
6X
7
X
8X
9X
10 X
Question 1
x
1X
2X
3X
4X
5X
6X
7X
8X
9X
10X
1
2
16
3
729
4
16
5
6
729
7
8
4
2
Why is 2 equivalent to 4 ? Write the values out in expanded
form and see if you can explain why.
Question 2
x
1X
2X
3X
4X
5X
6X
7X
8X
9X
10 X
1
2
16
3
64
4
5
1024
6
7
8
16 x 64 = 1024
2
3
5
4 x4 =4
Write the equivalent expressions in expanded form. Attempt to create a
rule for multiplying exponents with the same base.
Question 3
x
1X
2X
3X
8
27
4X
5X
6X
7X
8X
9X
10 X
1
2
3
216
4
5
6
7
8
8 x 27 = 216
3
3
3
2 x3 =6
Write the equivalent expressions in expanded form. Attempt to create a
rule for multiplying exponents with the same power.
Question 4
x
1X
2X
3X
4X
5
X
6
X
7
X
8
X
9
X
10
X
1
25
2
3
625
4
5
15625
6
7
8
15625 ÷ 625 = 25
6
4
2
5 ÷5 =5
Write the equivalent expressions in expanded form. Attempt to create a
rule for dividing exponents with the same base.
A. 1. Explain why each of the following statements is true.
3
2
5
A. 2 x 2 = 2
(2 x 2 x 2) x (2 x 2) = (2 x 2 x 2 x 2 x 2)
4
3
7
3
5
8
B. 3 x 3 = 3
C. 6 x 6 = 6
m
n
m+n
a xa =a
B. 1. Explain why each of the following statements is true.
3
3
3
A. 2 x 3 = 6
(2 x 2 x 2) x (3 x 3 x 3) = (2 x 3)(2 x 3)(2 x 3)
3
3
3
B. 5 x 6 = 30
4
4
4
C. 10 x 4 = 40
m
m
a x b = (ab)
m
C. 1. Explain why each of the following statements is true.
2
2 2
4
2
2 2
4
A. 4 = (2 ) = 2
B. 9 = (3 ) = 3
2
3 2
6
C. 125 = (5 ) = 5
m n
mn
(a ) = a
D. 1. Explain why each of the following statements is true.
A.
5
3
2
3
6
B.
4
5
4
3
= 3
1
= 4
m
a
C.
10
5
10
5
0
= 5
m-n
n
a
= a
Operating with Exponents
Examples
m
n
m
m
m+n
m
a x b = (ab)
mn
(a ) = a
2
2
6
2
5 x 3 = 15
3 2
6
(4 ) = 4
5
m
a
n
a
4
3 x3 =3
a xa =a
m n
2
m-n
=a
3
3
3
=3
2
84
3
5
Simplify: 4 x 4
15
A
4
B
4
C
4
D
47
8
2
85
7
3
Simplify: 5 ÷ 5
2
A
5
B
5
C
5
D
5
4
21
10
86
7
7
Simplify: 4 x 5
A
B
C
D
87
Simplify:
A
B
C
D
88
Simplify:
A
B
C
D
89
2 3
2
The expression (x z )(xy z) is equivalent to
A
2 2 3
xyz
B
3 3 4
C
xyz
D
xyz
4 2 5
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
90
Simplify:
A
B
C
D
91
Simplify:
A
B
C
D
simplified
92
The expression
is equivalent to
5
A
2w
B
2w
C
20w
D
20w
8
5
8
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
93
2
If x = - 4 and y = 3, what is the value of x - 3y ?
A
-13
B
-23
C
-31
D
-85
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
94
5
3
When -9 x is divided by -3x , x ≠ 0, the quotient is
2
A
–3x
B
3x
C
–27x
D
27x
2
15
8
From the New York State Education Department. Office of Assessment Policy, Development
and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra;
accessed 17, June, 2011.
By definition:
-1
x =
, x
0
95
-4
Which expression is equivalent to x ?
A
4
B
x
C
-4x
D
0
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
96
-3
What is the value of 2 ?
A
B
C
-6
D
-8
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
97
-1
2
Which expression is equivalent to x • y ?
A
B
xy2
C
D
-2
xy
From the New York State Education Department. Office of Assessment Policy,
Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Solving Equations with Perfect
Square and Cube Roots
Return to Table of
Contents
The product of two equal factors is the "square" of the number.
The product of three equal factors is the "cube" of the number.
When we solve equations, the solution sometimes requires finding a
square or cube root of both sides of the equation.
When your equation simplifies to:
2
x =#
you must find the square root of both sides in order to find the value of x.
When your equation simplifies to:
3
x =#
you must find the cube root of both sides in order to find the value of x.
Example:
Solve.
Divide each side by the coefficient.
Then take the square root of each side.
Example:
Solve.
Multiply each side by nine, then take
the cube root of each side.
Try These:
Solve.
± 10
±8
±9
±7
Try These:
Solve.
2
1
4
5
98
Solve.
99
Solve.
100
Solve.
101
Solve.
New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative (PMI)
For additional NJCTL Math content, visit
http://njctl.org/courses/math/.
Progressive Science Initiative (PSI)
For NJCTL Science content, visit http://njctl.org/courses/science/.
eInstruction
For information about Insight 360™ Classroom Instruction System,
visit http://www.einstruction.com.
For additional content samples, click here.