8th grade math numerical roots & radicals 2012-12-03
TRANSCRIPT
8th Grade Math
Numerical Roots & Radicals
www.njctl.org
2012-12-03
Numerical Roots and Radicals
Approximating Square Roots
Rational & Irrational Numbers
Radical Expressions Containing Variables
Simplifying Non-Perfect Square Radicands
Simplifying Perfect Square Radical Expressions
Simplifying Roots of Variables
Click on topic to go to that
section.
Squares, Square Roots & Perfect Squares
Squares of Numbers Greater than 20
Properties of Exponents
Solving Equations with Perfect Square & Cube Roots
Common Core Standards: 8.NS.1-2; 8.EE.1-2
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...
A = 42 = 4 4
= 16 sq.units
Area of a Square
The area (A) of a square can be found by squaring its side length, as shown below:
Click to see if the answer found with the Area formula is
correct!
A = s2
4 units
The area (A) of a square is labeled as
square units, or units2, because you
cover the figure with squares...
1 What is the area of a square with sides of 5 inches?
A
B
D
C
16 in2
25 in2
20 in2
30 in2
2 What is the area of a square with sides of 6 inches?
16 in2
24 in2
20 in2
36 in2
A
B
D
C
3 If a square has an area of 9 ft2, what is the
length of a side?
A
B
D
C
2 ft
3 ft
2.25 ft
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.
52 = 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?
What is nine squared?
49
81
Make a list of the numbers 1-15 and then square each of them.
Your paper should be set up as follows:
Number Square 1 1 2 4 3
(and so on)
Number Square
1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225
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 Square
1 12 43 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 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 Perfect Root Square
1 12 43 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 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 What is 132?
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.
102 = 100
202 = 400
302 = 900
402 = 1600
502 = 2500
602 = 3600
702 = 4900
802 = 6400
902 = 8100
1002 = 10000
What pattern do you notice?
For larger numbers, determine between which two multiples of ten the number lies.
102 = 100 1
2 = 1
202 = 400 2
2 = 4
302 = 900 3
2 = 9
402 = 1600 4
2 = 16
502 = 2500 5
2 = 25
602 = 3600 6
2 = 36
702 = 4900 7
2 = 49
802 = 6400 8
2 = 64
902 = 8100 9
2 = 81
1002 = 10000 10
2 = 100
Next, look at the ones digit to determine the ones digit of your square root.
2809
Examples:
Lies between 2500 & 3600 (50 and 60)Ends in nine so square root ends in 3
or 7Try 53 then 57
532 = 2809
Lies between 6400 and 8100 (80 and 90)
Ends in 4 so square root ends in 2 or 8Try 82 then 88
822 = 6724 NO!
882 = 7744
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?
12 =
8
2 = 15
2 =
22 = 9
2 = 16
2 =
32 = 10
2 = 17
2 =
42 = 11
2 = 18
2 =
52 = 12
2 = 19
2 =
62 = 13
2 = 20
2 =
72 = 14
2 =
Square Root Of A Number
Recall: If b2 = a, then b is a square root of a.
Example: If 42 = 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: = 4
Negative square root:- = - 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:
?
is not real
Evaluate the expression
24
?25
= ?26
27
28
= ?29
A
B
C
3
No real roots
-3
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.
A
B
D
C
-10
16
64
4
30 The expression equal to is equivalent to a positive integer when b is
Square Roots of Fractions
ab
= b 0
1649
= = 4
7
Try These
31
A
B D
C
no real solution
32
A
B D
C
no real solution
33
A
B D
C
no real solution
A
B D
C
no real solution
34
A
B D
C
no real solution
35
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.
= .05
= .2
= .3
Evaluate 36
A
B D
C
no real solution
Evaluate 37
A
C D
B.06 .6
6 No Real Solution
Evaluate 38
A
C D
B.11 11
1.1 No Real Solution
Evaluate 39
A
C D
B.8 .08
No Real Solution
Evaluate 40
A
C D
B
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?
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 Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225
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 Perfect Root Square
1 12 43 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 225
Estimate the following:
30
200
215
Square Perfect Root Square
1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225
Approximating a Square Root
Approximate to the nearest integer
< <
< <6 7
Identify perfect squares closest to 38
Take square root
Answer: Because 38 is closer to 36 than to 49, is closer to 6 than to 7. So, to the nearest integer, = 6
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.
Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8
2 2.2 2.4 2.6 2.8 3.0 2.1 2.3 2.5 2.7 2.9
√8
Example:
Approximate
10 10.2 10.4 10.6 10.8 11.0 10.1 10.3 10.5 10.7 10.9
A
B
D
C
9 and 16
36 and 49
25 and 36
49 and 64
41 The square root of 40 falls between which two perfect squares?
Identify perfect squares closest to 40
Take square root
Identify nearest integer
< <
<<
42 Which whole number is 40 closest to?
A
B
D
C
36 and 49
64 and 84
49 and 64
100 and 121
43 The square root of 110 falls between which two perfect squares?
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.
9649 Approximate to the nearest integer.
50 Approximate to the nearest integer.167
51 Approximate to the nearest integer.140
52 Approximate to the nearest integer. 40
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.
A
B
D
C
3 and 9
9 and 10
8 and 9
46 and 47
53 The expression is a number between
Rational & IrrationalNumbers
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.
Rational Irrational
25 36 10064 200
625
24
1225
0
3600
4032 52 225
300 1681
A BRational Irrational
54 Rational or Irrational?
A BRational Irrational
55 Rational or Irrational?
A BRational Irrational
56 Rational or Irrational?
A BRational Irrational
57 Rational or Irrational?
A BRational Irrational
58 Rational or Irrational?
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.
A
B
D
C
p
59 Which is a rational number?
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
D
C 3
2
4
Radical Expressions Containing Variables
Return to Table of Contents
To take the square root of a variable rewrite its exponent as the square of a power.
Square Roots of Variables
=
=
= x12
(a8)
2
= a8
(x12)2
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.
Square Roots of Variables
By Definition...
Examples
Try These.
= |x|5
= |x|13
no no
How many of these expressions will need an absolute value sign when simplified?
yes
yes
yes
yes
A
B
D
C
61 Simplify
A
B
D
C
62 Simplify
A
B
D
C
63 Simplify
A
B
D
C
64 Simplify
no real solution
A
B D
C
65
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
already in simplified form
A
B
D
C
66 Simplify
already in simplified form
A
B
D
C
67 Simplify
already in simplified form
A
B
D
C
68 Simplify
69 Simplify
already in simplified form
A
B
D
C
70 Simplify
already in simplified form
A
B
D
C
71 Simplify
already in simplified form
A
B
D
C
Which of the following does not have an irrational simplified form?
72
A
B
D
C
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
D
C
74
A
B
D
C
Simplify
75
A
B
D
C
Simplify
76
A
B
D
C
Simplify
77
A
B
D
C
Simplify
20
10
7
4
78
A
B
D
C
When is written in simplest radical form, the
result is . What is the value of k?
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 simplestform, what is the value of a?
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.
6
2
3
8
A
B
D
C
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
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.
Simplify
80
A
B
D
C
Simplify
A
B
D
C
81 Simplify
A
B
D
C
82 Simplify
A
B
D
C
83 Simplify
Properties of Exponents
Return to Table of Contents
Rules of Exponents
Exponential Table Questions.pdf
Exponential Table.pdf
Exponential Test Review.pdf
Materials
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 9 10
1
2
3
4
5
6
7
8
X X X X X X X X XX
Why is 24 equivalent to 4
2? Write the values out in
expanded form and see if you can explain why.
Question 1
x 1 2 3 4 5 6 7 8 9 10
1
2 16
3 72
9
4 16
5
6 72
9
7
8
X X X X X X X X XX
16 x 64 = 1024 42 x 4
3 = 4
5
Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same base.
Question 2
x 1 2 3 4 5 6 7 8 9 10
1
2 16
3 64
4
5 102
4
6
7
8
X X X X X X X X XX
8 x 27 = 216 23 x 3
3 = 6
3
Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same power.
Question 3
x 1 2 3 4 5 6 7 8 9 10
1
2
3 8 27 21
6
4
5
6
7
8
X X X X X X X X XX
15625 ÷ 625 = 25 56 ÷ 5
4 = 5
2
Write the equivalent expressions in expanded form. Attempt to create a rule for dividing exponents with the same base.
Question 4
x 1 2 3 4 5 6 7 8 9 10
1
2 25
3
4 625
5
6 1562
5
7
8
X X X X X X X X XX
A. 1. Explain why each of the following statements is true.
A. 23 x 2
2 = 2
5
(2 x 2 x 2) x (2 x 2) = (2 x 2 x 2 x 2 x 2)
B. 34 x 3
3 = 3
7
C. 63 x 6
5 = 6
8
am x a
n = a
m + n
B. 1. Explain why each of the following statements is true.
A. 23 x 3
3 = 6
3
(2 x 2 x 2) x (3 x 3 x 3) = (2 x 3)(2 x 3)(2 x 3)
B. 53 x 6
3 = 30
3
C. 104 x 4
4 = 40
4
am x b
m = (ab)
m
C. 1. Explain why each of the following statements is true.
A. 42 = (2
2)
2 = 2
4
B. 92 = (3
2)
2 = 3
4
C. 1252 = (5
3)
2 = 5
6
(am)
n = a
mn
D. 1. Explain why each of the following statements is true.
A. 35
32
46
B. 45
C. 510
510
=
=
=
33
41
50 a
m
an
= am-n
Operating with Exponents
32 x 3
4 = 3
6
52 x 3
2 = 15
2
am x a
n = a
m+n
am x b
m = (ab)
m
(43)
2 = 4
6
Examples
(am)
n = a
mn
am
= am-n
an
35
= 32
33
415
48
42
47
84
A
B
D
C
Simplify: 43 x 4
5
52
54
521
510
85
A
B
D
C
Simplify: 57 ÷ 5
3
86
A
B
D
C
Simplify: 47 x 5
7
87
A
B
D
C
Simplify:
88
A
B
D
C
Simplify:
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.
x2y
2z
3
x3y
3z
4
x4y
2z
5
89
A
B
D
C
The expression (x2z
3)(xy
2z) is equivalent to
90
A
B
D
C
Simplify:
simplified
91
A
B
D
C
Simplify:
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.
2w5
2w8
20w5
20w8
92
A
B
D
C
The expression is equivalent to
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.
-13
-23
-31
-85
93
A
B
D
C
If x = - 4 and y = 3, what is the value of x - 3y2?
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.
–3x2
3x2
–27x15
27x8
94
A
B
D
C
When -9 x5 is divided by -3x
3, x ≠ 0, the quotient is
By definition:
x-1 = , x 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.
x4
-4x
0
95
A
B
D
C
Which expression is equivalent to x-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.
-6
-8
96
A
B
D
C
What is the value of 2-3?
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
A
B D
C
Which expression is equivalent to x-1• y
2?
xy2
xy-2
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:
x2 = #
you must find the square root of both sides in order to find the value of x.
When your equation simplifies to:
x3 = #
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.