Transcript Module8_Lesson2_Simplify Radical Expressions Remediation

Radicals
Table of Contents
 Slides 3-13: Perfect Squares
 Slides 15-19: Rules
 Slides 20-22: Simplifying Radicals
 Slide 23: Product Property
 Slides 24-31: Examples and Practice Problems
 Slides 32-35: Perfect Cubes
 Slides 36-40: Nth Roots
 Slides 41-48: Examples and Practice Problems
 Slides 49-53: Solving Equations
Audio/Video and Interactive Sites
 Slide 14: Gizmos
 Slide 19: Gizmo
 Slide 24: Gizmo
 Slide 27: Gizmo
 Slide 48: Interactive
What are Perfect Squares?
1•1=1
2•2=4
3•3=9
4 • 4 = 16
5 • 5 = 25
6 • 6 = 36
49, 64, 81, 100, 121, 144, ...
and so on….
Since 42  16 , 4  16 .
 The symbol,
, is called a radical sign.
 Finding the square root of a number and squaring a number are inverse
operations.
 To find the square root of a number n, you must find a number whose square is
n. For example,
49 is 7, since 72 = 49.
Likewise, (–7)2 = 49, so –7 is also a square root of 49.
We would write the final answer as:
49  7
 An expression written with a radical sign is called a radical expression.
 The expression written under the radical sign is called the radicand.
NOTE: Every positive real number has two real number square
roots.
169  13
The number 0 has just one square root, 0 itself.
0 0
Negative numbers do not have real number square roots.
 4  No Real Roots
When evaluating we choose the positive value of a called
the principal root.
Evaluate
169
 13
Notice, since we are evaluating, we only
use the positive answer.
For any real numbers a and b,
if a2 = b,
then a is a square root of b.
a 2  b then
7 2  49 then
112  121 then
b a
49  7
121  11
Just like adding and subtracting are inverse operations,
finding the square root of a number and squaring a number are inverse operations.
Perfect Square
2x2=4
2
The square root of 4 is ...
2
2
4 2
Perfect Square
3x3=9
The square root of 9 is ...
3
3
3
9 3
Perfect Square
The square root of 16 is ...
4 x 4 = 16
4
4
4
16  4
Perfect Square
5
5 x 5 = 25
5
Can you guess what the square root of 25 is?
The square root of 25 is ...
5
25  5
This is great,
But….
Do you really want to draw blocks for a problem like…
2
11
probably not!
If you are given a problem like this:
Find 2025
Are you going to have fun getting this answer by drawing
2025 blocks? Probably not!!!!!!
2025  45
It is easier to memorize the perfect squares up to a certain
point. The following should be memorized. You will see them
time and time again.
x
x2
x
x2
0
0
10
100
1
1
11
121
2
4
12
144
3
9
13
169
4
16
14
196
5
25
15
225
6
36
16
256
7
49
20
400
8
64
25
625
9
81
50
2500
Gizmo: Ordering and Approximating Square Roots
Gizmo: Ordering and Approximating Square Roots
Quick Facts about Radicals
 a  b
To name the negative square root of a, we say
 25  5
 a  b
To indicate both square roots, use the plus/minus
sign which indicates positive or negative.
 25  5
3
4
7
x
1
x2
x
1
x3
x
1
x4
x
1
x7
n
x
1
n
x
Simplifying Radicals
 a  No Real Solution
• Negative numbers do not have real number
square roots.
• No Real Solution
 25  No Real Solution
a =b
 This symbol represents the principal square root of a.
 The principal square root of a non-negative number is
its nonnegative square root.
25  5
Gizmo: Square Roots
Simplifying Radicals
 8 99 x5 y 3 z 2
Divide the number
under the radical.
If all numbers are not
prime, continue
dividing.
Find pairs, for a square
root, under the radical
and pull them out.
Multiply the items you
pulled out by anything
in front of the radical
sign.
 8 9 11 x  x  x  x  x  y  y  y  z  z
 8 3  3 11 x  x  x  x  x  y  y  y  z  z
 8 3  3  11  x  x  x  x  x  y  y  y  z  z
3
Multiply anything left
under the radical .
x
x
y
 8  3  x  x  y  z 11xy
It is done!
 24x2 yz 11xy
z
Evaluate the following:
81
99  9
To solve:
Find all factors
Pull out pairs (using one number to
represent the pair. Multiply if
needed)
100
0.25
1
4
0.5  0.5  0.5
1 1 1
 
2 2 2
2255
25
 10
x6
x3  x3  x3
100
10  10
 10
99  9
Find all real roots:
81
 9  9  9
0.5  0.5  0.5
0.25
1
4
81  9
 0.5  0.5  0.5
1 1 1
 
2 2 2
1
1
1
   
2
2
2
0.25  0.5
1
1

4
2
Not all numbers are perfect squares
• To find the roots, you will need to simplify
radial expressions in which the radicand is not
a perfect square using the Product Property of
Square Roots.
ab  a  b
THIS IS WHERE KNOWING THE PERFECT SQUARES IS VITAL
x
x2
x
x2
0
0
10
100
1
1
11
121
2
4
12
144
3
9
13
169
4
16
14
196
5
25
15
225
6
36
16
256
7
49
20
400
8
64
25
625
9
81
50
2500
Gizmo: Simplifying Radicals
Examples:
A. Simplify 50
Steps
Explanation
50  5  5  2
 25  2
Prime Factorizat ion
(5)(5)  25 - A Perfect Square
5 2
Simplify
25
B. Simplify
147
Steps
Explanation
147  7  7  3
 49  3
Prime Factorizat ion
(7)(7)  49 - A Perfect Square
7 3
Simplify
49
xy  x y
The general rule for reducing the radicand is to remove
any perfect powers.
We are only considering square roots here, so what we
are looking for is any factor that is a perfect square.
In the following examples we will assume that x is
positive.
Gizmo: Simplifying Radicals
Examples:
A. Evaluate
16 x
16 x  16 x  4  4 x  4 x
B. Evaluate
x3
x  x x x
3
2
2
xx x
Although x 3 is not a perfect square, it has a factor of x 2 ,
which is the square of x.
Examples:
C. Evaluate
x5 
x5
x4 x 
x4
x
x2  x2
x  x2 x
Here the perfect square factor is x 4 , which is the
square of x 2 .
D. Evaluate
8x5
8x5  4  2  x 4  x  4 x 4 2 x  2 x 2 2 x
In this example we could take out a 4 and a factor
of x 2 , leaving behind a 2 and one factor of x.
Examples:
E.
3
80 x 7 y 3 
 3 2  2  2  2  5  xxxxxxx yyy
 3 (2  2  2)  2  5  ( xxx)( xxx) x  ( yyy)
*All the sets of “3” have been grouped. They are cubes!
 2x2 y
 2x2 y
25 x
3
3
10 x
Unless otherwise stated, when simplifying expressions using
variables, we must use absolute value signs.
n
an  a
when n is even.
NOTE:
No absolute value signs are needed when finding cube roots, because a real number
has just one cube root. The cube root of a positive number is positive. The cube root
of a negative number is negative.
Evaluate the following:
 16
No real roots
49x 4
7  7  x2  x2  7 x2
1 3 2

x y
8
 9 255m
1 1
1

 x  x  x  y  y   xy x
4 4
4
9
 9 5  51  m2  m2  m2  m2  m
 9m2m2 255m
 9m 4 255m
What are Cubes?
• 13 = 1 x 1 x 1 = 1
• 23 = 2 x 2 x 2 = 8
• 33 = 3 x 3 x 3 = 27
• 43 = 4 x 4 x 4 = 64
• 53 = 5 x 5 x 5 = 125
• and so on and on and on…..
Cubes
5
7
6
8
1
3
2
4
2x2x2=8
2
2
2 8
3
2
3 x 3 x 3 = 27
3
3
3  27
3
3
th
N
Roots
When there is no index number, n, it is
understood to be a 2 or square root.
For example:
x
= principal square root of x.
Not every radical is a square root.
If there is an index number n other than the
number 2, then you have a root other than a
square root.
th
N
Roots
• Since 32 = 9. we call 3 the square root of 9.
9 3
• Since 33 =27 we call 3 the cube root of 27.
3
27  3
• Since 34 = 81, we call 3 the fourth root of 81.
4
81  3
• This leads us to the definition of the nth
root of a number. If an = b then a is
the nth root b notated as, a  n b .
More Explanation of Roots
th
N
Roots
• Since (-)(-) = + and (+)(+) = + , then all positive
real numbers have two square roots.
• Remember in our Real Number System the  b
is not defined.
• However we can find the cube root of
negative numbers since (-)(-)(-) = a negative
and (+)(+)(+) = a positive.
• Therefore, cube roots only have one root.
Nth Roots
Type of Number
+
Number of Real
nth Roots when
n is even
2
Number of Real
nth Roots when
n is odd.
1
0
1
1
-
None
1
Nth Roots of Variables
• Lets use a table to see the pattern when
simplifying nth roots of variables.
x x
x2
3
3
n
x
4
x
3
x
6
x
m
x  x  x x
2
3
3
2
 x
 x2
x x x
x
x  x  x x
 x2
3
3
x
m
n
*Note: In the first row above, the absolute value of x yields the principal
root in the event that x is negative.
Examples:
A. Find all real cube roots of -125, 64, 0 and 9.
Solutions : - 5, 4, 0 and 3 9
B. Find all real fourth roots of 16, 625, -1 and 0.
Solutions :  2,  5, Undefined and 0
As previously stated when a number has two real roots, the
positive root is called the principal root and the Radical
indicates the principal root. Therefore when asked to find the
nth root of a number we always choose the principal root.
F. Simplfy 3 1000 x 3 y 9
3
1000 x y  10 x ( y )
3 9
3
3 3
3 3
 3 (10 xy3 ) 3
 10xy3
Write each factor as a
cube.
Write as the cube of
a product.
Simplify.
Absolute Value signs are NOT needed here because the
index, n, is odd.
Application/Critical Thinking
4
A.
The formula for the volume of a sphere is V  3 . r
Find the radius, to the nearest hundredth, of a sphere with a volume of
. in 3
15
B.
A student visiting the Sears Tower Skydeck is 1353 feet above the
ground. Find the distance the student can see to the horizon. Use the
formula d  1.5h to the approximate the distance d in miles to the
horizon when h is the height of the viewer’s eyes above the ground in
feet. Round to the nearest mile.
C.
2
A square garden plot has an area of 24 ft.
a. Find the length of each side in simplest radical form.
b. Calculate the length of each side to the nearest tenth of a foot.
3
Application Solutions:
A.
4
15   r 3
3
3
3 4

(15)    r 3 
4
4 3

45
  r3
4
11.25  3.14r 3
11.25
r
3.14
r  1.53in
3
B. d  1.5h
d  (1.5)(1353)
d  2029.5
d  45miles
C. A  s 2
24  s
s
2
24
a) s  2 6
b) 4.9 ft
Evaluate the following:
3
3
64
88  3 2  4 2  4
To solve:
Find all factors
Pull out set’s that contain the same number of
terms as the root (using one number to
represent the set of 4. Multiply if needed)
 3 222222
 22
4
4
4
4
4
4
16
44
x
4
99
4
xxxx
x
3
1000
3 10 10 10
3333
 4 2222
2
4
81
 10
3
Evaluate the following:
3

 6  6  6
3
 216
 6
1
1

144 2    144  12


 144 2




1
(144) 2
5
32
16807
No real roots
5
25
2

5
7
7
5
 486m 20n15 p3
 5 (243  2)m 20n15 p3
 5 (3)5  2m 20n15 p3
 3m 4 n3 5 2 p 3
Practice Problems and Answers