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 = 12 • 2 = 43 • 3 = 9
4 • 4 = 165 • 5 = 256 • 6 = 36
49, 64, 81, 100, 121, 144, ...and so on….
Since , . 1642 164
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,
is 7, since 72 = 49.49
Likewise, (–7)2 = 49, so –7 is also a square root of 49.
We would write the final answer as:We would write the final answer as: 749
The symbol, , is called a radical sign.
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.
The number 0 has just one square root, 0 itself.
Negative numbers do not have real number square roots.
When evaluating we choose the positive value of a called the principal root.
13169
00
RootsRNo eal 4
Evaluate 169 13Notice, since we are evaluating, we only use the positive answer.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.
abthenba 2
749 4972 then
11121 121112 then
Just like adding and subtracting are inverse operations, finding the square root of a number and squaring a number are inverse operations.
Just like adding and subtracting are inverse operations, finding the square root of a number and squaring a number are inverse operations.
2
2
2 x 2 = 4
Perfect Square
The square root of 4 is ...
2
24
3 x 3 = 9
3
3
Perfect Square
The square root of 9 is ...
3
39
4 x 4 = 16
4
4
Perfect Square
4The square root of 16 is ...
416
5
5
5 x 5 = 25
Perfect Square
Can you guess what the square root of 25 is?
5The square root of 25 is ...
525
This is great, But….
Do you really want to draw blocks for a problem like…
probably not!
211
If you are given a problem like this:
Find
Are you going to have fun getting this answer by drawing 2025 blocks? Probably not!!!!!!
2025
452025
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
ba
To name the negative square root of a, we say
525
ba To indicate both square roots, use the plus/minus sign which indicates positive or negative.
525
7
17
4
14
3
13
2
1
xx
xx
xx
xx
nn xx1
Simplifying Radicals
• Negative numbers do not have real number square roots.
• No Real Solution
Solution Real No a
Solution Real No25
= b
This symbol represents the principal square root of a.
The principal square root of a non-negative number is its nonnegative square root.
a
525
Gizmo: Square Roots
Simplifying Radicals235998 zyx
zzyyyxxxxx 1198Divide the number under the radical.If all numbers are not prime, continue dividing.
zzyyyxxxxx 11338
Find pairs, for a square root, under the radical and pull them out.
zyxx
zzyyyxxxxx
3
11338
Multiply the items you pulled out by anything in front of the radical sign.
Multiply anything left under the radical . xyzyxx 1138
xyyzx 1124 2It is done!
Evaluate the following:
81
25.0
4
1
999
5.05.05.0
2
1
2
1
2
1
6x 333 xxx
To solve: Find all factorsPull out pairs (using one number to represent the pair. Multiply if needed)
100
5522
52
10
100
1010
10
Find all real roots:
81
25.0
4
1
999
5.05.05.0
2
1
2
1
2
1
999
5.05.05.0
2
1
2
1
2
1
981
5.025.0
2
1
4
1
• 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.
baab
Not all numbers are perfect squaresNot all numbers are perfect squares
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
25
225
25550
25Simplify
SquarePerfect A - 25(5)(5)
ionFactorizat Prime
B. Simplify 147
Steps Explanation
37
349
377147
49Simplify
SquarePerfect A - 49(7)(7)
ionFactorizat Prime
yxxy
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
B. Evaluate
xxxx 4441616
xxxxxxx 223
. xof square theiswhich
, xoffactor a hasit square,perfect anot is Although x 23
x16
3x
xxxxxxxxxx 222445
. xof square
theis which , xisfactor squareperfect theHere2
4
xxxxxxx 2224248 2445
x.offactor one and 2 a behind leaving , xof
factor a and 4 aout takecould weexample In this2
Examples:
C. Evaluate
D. Evaluate
5x
58x
Examples:
E.
3 3780 yx3 52222 yyyxxxxxxx3 )())((52)222( yyyxxxxxxx
32 52 2 xyx 32 10 2 xyx
Unless otherwise stated, when simplifying expressions using variables, we must use absolute value signs.
aan n when n is even.
*All the sets of “3” have been grouped. They are cubes!
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
449x
23
8
1yx
92559 m
No real roots
222 777 xxx
xxyyyxxx4
1
4
1
4
1
mmmmm 22225159
mmm 2559 22 mm 2559 4
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…..
1 23 4
5 67 8
Cubes
2
2
2
2 x 2 x 2 = 8
823
3 x 3 x 3 = 27
2733
3
3
3
Nth Roots
When there is no index number, n, it is understood to be a 2 or square root.
For example: = 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.
x
• Since 32 = 9. we call 3 the square root of 9.
• Since 33 =27 we call 3 the cube root of 27.
• Since 34 = 81, we call 3 the fourth root of 81.
Nth Roots
39
3273
3814
More Explanation of Roots
n ba
• 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, .
Nth Roots
• Since (-)(-) = + and (+)(+) = + , then all positive real numbers have two square roots.
• Remember in our Real Number System the 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.
b
Nth Roots
Type of Number Number of Real nth Roots when
n is even
Number of Real nth Roots when
n is odd.+ 2 1
0 1 1
- None 1
Nth Roots of Variables• Lets use a table to see the pattern when
simplifying nth roots of variables.2x
4x
3 3x
3 6x
n mx
xx
xxxx 22
3 xxx
xxxx 3 33
x
2xx2xn
mx
*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.
B.Find all real fourth roots of 16, 625, -1 and 0.
3 9 and 0 4, 5,- :Solutions
0 and Undefined5, 2, : Solutions
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. 3 931000 yxSimplfy
3 33333 93 )(101000 yxyx Write each factor as a cube.
3 33)10( xy Write as the cube of a product.
310xy Simplify.
Absolute Value signs are NOT needed here because the index, n, is odd.
Application/Critical Thinking
A. The formula for the volume of a sphere is . Find the radius, to the nearest hundredth, of a sphere with a volume of
.
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 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. A square garden plot has an area of .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 3
4rV
3in 15
hd 5.1
2 24 ft
Application Solutions:
A. B.
inr
r
r
r
r
r
53.114.3
25.11
14.325.11
4
45
3
4
4
3)15(
4
3
3
415
3
3
3
3
3
milesd
d
d
hd
45
5.2029
)1353)(5.1(
5.1
C.
ftb
sa
s
s
sA
9.4 )
62 )
24
24 2
2
Evaluate the following:
4 162
2222
444
4
To solve: Find all factorsPull 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)
814
994
3
3 1000
1010103
10
33334
4 4xx
4 xxxx
3 64
4
22
222222
4242883
33
3 2166
6663
Evaluate the following:
2
1
144 121441442
1
2
1
)144( No real roots
516807
32
7
2
7
255
5
5 31520486 pnm
5 31520)2243( pnm
5 315205 2)3( pnm
5 334 2 3 pnm
Practice Problems and Answers
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