Introduction to Radicals

1
Introduction to Radicals
2
Square Root of a Number
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
3
Radical Expressions
Finding a root of a number is the inverse
operation of raising a number to a power.
radical sign
index
radicand
This symbol is the radical or the radical sign

• The expression under the radical sign is the
  radicand.

• The index defines the root to be taken.

4
Terminology

• 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.

5
Square Roots
A square root of any positive number has two
roots one is positive and the
other is negative.
If a is a positive number, then
is the positive (principal) square root of a and
is the negative square root of a.
Examples
non-real
6
What does the following symbol represent?
The symbol represents the positive or principal
root of a number.
What is the radicand of the expression ?
5xy
7
What does the following symbol represent?
The symbol represents the negative root of a
number.
What is the index of the expression ?
3
8
What numbers 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, ...
9
Perfect Squares
64
225
1
81
256
4
100
289
9
121
16
324
144
25
400
169
36
196
49
625
10
Simplify
2
4
5
This is a piece of cake!
10
12
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Simplifying Radicals
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Simplifying Radical Expressions
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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.

14
Perfect Square Factor Other Factor
Simplify




LEAVE IN RADICAL FORM






15
Perfect Square Factor Other Factor
Simplify




LEAVE IN RADICAL FORM






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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

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Simplify
Square root of a variable to an even power the
variable to one-half the power.
18
Simplify
Square root of a variable to an even power the
variable to one-half the power.
19
Simplify
20
Simplify
21
Simplify

1. .
2. .
3. .
4. .

22
Simplify

1. 3x6
2. 3x18
3. 9x6
4. 9x18

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Combining Radicals

To combine radicals combine the coefficients of
like radicals
24
Simplify each expression
25
Simplify each expression Simplify each radical
first and then combine.
26
Simplify each expression Simplify each radical
first and then combine.
27
Perfect Square Factor Other Factor
Simplify




LEAVE IN RADICAL FORM






28
Simplify each expression
29
Simplify each expression
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Homework radicals 1

• Complete problems 1-24 EVEN from worksheet

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Multiplying Radicals

To multiply radicals multiply the coefficients
and then multiply the radicands and then simplify
the remaining radicals.
32
Multiply and then simplify
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(No Transcript)
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Dividing Radicals
To divide radicals divide the coefficients,
divide the radicands if possible, and rationalize
the denominator so that no radical remains in the
denominator
35
That was easy!
36
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.
42 cannot be simplified, so we are finished.
37
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.
38
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.
Reduce the fraction.
39
Simplify
X
Y3
P2X3Y
2X2Y
5C4D5
40
Simplify




41
Homework
Classwork Packet in Yellow Folder under the
desk --- 2nd page
worksheet --- Non-Perfect Squares (1-12)
42
Homework radicals 2

• Complete problems 1-15 from worksheet.