Introduction to Radicals
If b 2 = a, then b is a square root of a.
Meaning Positive
Square Root
Negative
Square Root
The positive and negative square
roots
Symbol
Example
39 39 39
Radical Expressions
Finding a root of a number is the inverse operation of raising a number to a power.
This symbol is the radical or the radical sign
n aindex
radical sign
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
• square root: one of two equal factors of a given number. The radicand is like the “area” of a square and the simplified answer is the length of the side of the squares.
• Principal square root: the positive square root of a number; the principal square root of 9 is 3.
• negative square root: the negative square root of 9 is –3 and is shown like
• radical: the symbol which is read “the square root of a” is called a radical.
• radicand: the number or expression inside a radical symbol --- 3 is the radicand.
• perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc… are perfect squares.
39
39
3
Square Roots
If a is a positive number, then
a is the positive (principal) square root of a and
100
a is the negative square root of a.
A square root of any positive number has two roots – one is positive and the other is negative.
Examples:
10
25
49
5
7
11 36 6
9 non-real # 81.0 9.0
What does the following symbol represent?
The symbol represents the positive or principal root of a number.
4 5xyWhat is the radicand of the expression ?
5xy
What does the following symbol represent?
The symbol represents the negative root of a number.
3 525 yxWhat is the index of the expression ?
3
What numbers are perfect squares?
1 • 1 = 12 • 2 = 43 • 3 = 9
4 • 4 = 16 5 • 5 = 25 6 • 6 = 36
49, 64, 81, 100, 121, 144, ...
Perfect Squares
1
4
916
253649
64
81
100121
144169196
225
256
324
400
625
289
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
Simplifying Radicals
Simplifying Radical Expressions
100 4 25
36 4 9
326
Product Property for Radicals
ab a b
10 2 5 36 4 9
Simplifying Radical Expressions
• A radical has been simplified when its radicand contains no perfect square factors.
• Test to see if it can be divided by 4, then 9, then 25, then 49, etc.
• Sometimes factoring the radicand using the “tree” is helpful.
Product Property for Radicals
50 25 2 5 2
14 7x x
8
20
32
75
40
=
= =
=
=
2*4
5*4
2*16
3*25
10*4
=
=
=
=
=
22
52
24
35
102
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
48
80
50
125
450
=
= =
=
=
3*16
5*16
2*25
5*25
2*225
=
=
=
=
=
34
54
25
55
215
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
Steps to Simplify Radicals:
1. Try to divide the radicand into a perfect square for numbers
2. If there is an exponent make it even by using rules of exponents
3. Separate the factors to its own square root
4. Simplify
Simplify:12x
6x
26x
Square root of a variable to an even power = the
variable to one-half the power.
Simplify:88y
44y
Square root of a variable to an even power = the
variable to one-half the power.
Simplify:13x
xx6
12x x
112xx
Simplify:750y
35 2y y
625 2y y
Simplify
1. .
2. .
3. .
4. .
2 18
72
3 8
6 236 2
Simplify 369x
1. 3x6
2. 3x18
3. 9x6
4. 9x18
+To combine radicals: combine the coefficients of like radicals
Simplify each expression
737576 78
62747365 7763
Simplify each expression: Simplify each radical first and then combine.
323502
22
212210
24*325*2
2*1632*252
Simplify each expression: Simplify each radical first and then combine.
485273
329
32039
34*533*3
3*1653*93
18
288
75
24
72
=
= =
=
=
=
=
=
=
=
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
Simplify each expression
636556
547243
32782
Simplify each expression
20556
32718
6367282
Homework radicals 1
• Complete problems 1-24 EVEN from worksheet
*To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.
35*5 175 7*25 75
Multiply and then simplify
73*82 566 14*46
142*6 1412
204*52 1008 8010*8
2
5 5*5 25 5
2
7 7*7 49 7
2
8 8*8 64 8
2
x xx * 2x x
To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator
7
56 8 2*4 22
7
6This cannot be
divided which leaves the radical in the
denominator. We do not leave radicals in the denominator. So
we need to rationalize by multiplying the
fraction by something so we can eliminate
the radical in the denominator.
7
7*
7
6
49
42
7
42
42 cannot be simplified, so we are
finished.
This can be divided which leaves the
radical in the denominator. We do not leave radicals in the denominator. So
we need to rationalize by multiplying the
fraction by something so we can eliminate
the radical in the denominator.
10
5
2
2*
2
1
2
2
This cannot be divided which leaves
the radical in the denominator. We do not leave radicals in the denominator. So
we need to rationalize by multiplying the
fraction by something so we can eliminate
the radical in the denominator.
12
3
3
3*
12
3
36
33
6
33
2
3Reduce
the fraction.
2X
6Y
264 YXP
244 YX
10825 DC
= X
= Y3
= P2X3Y
= 2X2Y
= 5C4D5
3X
XX
=
=
XX *2
YY 45Y
=
= YY 2
Homework:
worksheet --- Non-Perfect Squares (#1-12)
Classwork:
Packet in Yellow Folder under the desk --- 2nd page
Homework radicals 2
• Complete problems 1-15 from worksheet.
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