Download Chapter 8: Roots and Radicals

Warm-Up #5
1.
Find the product of ab.
Let a =
1
2
𝑎𝑛𝑑 𝑏 =
2.
Simplify 91
3.
Estimate: What is
25
.
36
17
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Homework
Advanced: Simplifying Radical Worksheet
Page 1. #1-6
Page 2. #1-6
Regular: Simplifying Radical Worksheet
Page 1. #1-4
Page 2. #1-4
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Introduction to Radicals
Principal Square Roots
The principal (positive) square root is noted
as
a
The negative square root is noted as
 a
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Perfect Squares
1
64
225
4
81
256
9
16
100
121
289
25
36
49
144
169
196
400
324
625
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16
= 4 or -4
25
= 5 or -5
100
144
= 10 or -10
= 12 or -12
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LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
4*5
=
2 5
20
=
32
=
16 * 2 =
4 2
75
=
25 * 3 =
5 3
40
=
4 *10 = 2 10
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Cube Roots
The cube root of a real number a
3
a  b only if b 3  a
Example:
3
8  2 because 23  (2)(2)(2)  8
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15.1 – Introduction to Radicals
Cube Roots
3
a
A cube root of any positive number is positive.
A cube root of any negative number is negative.
Examples:
3
3
27  3
 27   3
3
8  2
3
5
125
3

4
64
8 2
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Cube Roots
Example
3
3
27 
3
 8x   2x
6
2
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Simplifying Radicals
Product Rule for Radicals
If
a and b are real numbers,
ab  a  b
a
a

if
b
b
b 0
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Simplifying Radicals
Example
Simplify the following radical expressions.
40 
4  10  2 10
5

16
5
5

4
16
15
No perfect square factor, so the
radical is already simplified.
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Simplifying Radicals
Example
Simplify the following radical expressions.
x 
6
x x 
x  x x
20

16
x
20
4 5
2 5

8
x
x8
7
x16

6
3
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x
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Quotient Rule for Radicals
If n a and n b are real numbers,
n
n
ab  n a  n b
a na
 n if
b
b
n
b 0
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Simplifying Radicals
Example
Simplify the following radical expressions.
3
3
16  3 8  2 
3

64
3
3
3

64
3
3
8 3 2  2 3 2
3
4
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Adding and Subtracting
Radicals
Sums and Differences
Rules in the previous section allowed us to
split radicals that had a radicand which was a
product or a quotient.
We can NOT split sums or differences.
ab  a  b
a b  a  b
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Like Radicals
What is combining “like terms”?
Similarly, we can work with the concept of
“like” radicals to combine radicals with the
same radicand.
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Adding and Subtracting Radical Expressions
Example
37 3  8 3
10 2  4 2  6 2
3
2 4 2
Can not simplify
5 3
Can not simplify
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Adding and Subtracting Radical Expressions
Example
Simplify the following radical expression.
 75  12  3 3 
 25  3  4  3  3 3 
 25  3  4  3  3 3 
5 3  2 3 3 3 
 5  2  3
3  6 3
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Adding and Subtracting Radical Expressions
Example
Simplify the following radical expression.
3
64  3 14  9 
4  3 14  9   5  3 14
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Adding and Subtracting Radical Expressions
Example
Simplify the following radical expression. Assume
that variables represent positive real numbers.
3 45x3  x 5x  3 9 x 2  5x  x 5x 
3 9 x  5x  x 5x 
2
3  3x 5 x  x 5 x 
9 x 5x  x 5x 
9 x  x 
5x 
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10 x 5 x
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Multiplying and Dividing
Radicals
Multiplying and Dividing Radical Expressions
If
n
a and n b are real numbers,
n
a  n b  n ab
n
a n a

if b  0
b
b
n
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Multiplying and Dividing Radical Expressions
Example
Simplify the following radical expressions.
3 y  5x  15 xy
7 6
ab
3 2
ab

7 6
ab

3 2
ab
ab  ab
4 4
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Rationalizing the Denominator
If we rewrite the expression so that there is no
radical in the denominator, it is called
rationalizing the denominator.
Rationalizing the denominator is the process
of eliminating the radical in the denominator.
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Rationalizing the Denominator
Example
Rationalize the denominator.
3
2


2
2
6
3 2

2
2 2
3
6 33
63 3
63 3
6 3
3




 2 3
3
3
3
3
3
3
27
3
9 3
9
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Conjugates
Many rational quotients have a sum or difference
of terms in a denominator, rather than a single
radical.
32
32
2 3
5 1
• need to multiply by the conjugate of the
denominator
• The conjugate uses the same terms, but the
opposite operation (+ or ).
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Rationalizing the Denominator
Example
Rationalize the denominator.
2 3
3  2 3 2 2  2 3
32



2  3 2 2  3  2  3 3
2 3
6 3 2 2  2 3

23
6 3 2 2  2 3

1
 6 3 2 2  2 3
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