Chapter 8
Radical
Expressions and
Equations
Study Strategy – Note Taking
•Choosing a Seat
•Note-Taking Speed
•Things to Include
•Being an Active Learner
•Formatting Your Notes
•Rewriting Your Notes
•Write Down Each Step
Section 8.1 – Square Roots and Radical Notation
Concept – Radical Expressions, Square Roots
The principal square root of b, denoted b ,
is the positive number a such that a 2 = b.
The expression b is called a radical
expression.
The sign
is called a radical sign.
The expression contained inside the
radical sign is called the radicand.
Section 8.1 – Square Roots and Radical Notation
Example – Radical Expressions, Square Roots
Simplify: 49
49 = 7 because 7 = 49.
2
Section 8.1 – Square Roots and Radical Notation
Example – Radical Expressions, Square Roots
10
Simplify, assume x is nonnegative: 81x
81x = 9 x because ( 9 x
10
5
5
)
2
= 81x .
10
Section 8.1 – Square Roots and Radical Notation
Concept – Product Property of Radicals
For any index n > 1 and any nonnegative
n
n
n
numbers a and b, a ⋅ b = ab .
Section 8.1 – Square Roots and Radical Notation
Example – Product Property of Radicals
Simplify:
50 ⋅ 18
50 ⋅ 18 = 900
= 30
Section 8.1 – Square Roots and Radical Notation
Example – Product Property of Radicals
Simplify:
7 2 ⋅6 2
7 2 ⋅ 6 2 = 42 4
= 42 ⋅ 2
= 84
Section 8.1 – Square Roots and Radical Notation
Concept – Quotient Property of Radicals
For any index n > 1 and any nonnegative
number a and positive number b, n a n a .
n
b
=
b
Section 8.1 – Square Roots and Radical Notation
Example – Quotient Property of Radicals
Simplify:
396
11
396
= 36
11
=6
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Concept – Simplifying Radicals
1. Completely factor the radicand.
2. Rewrite each factor as a product of two
factors. The exponent for the first factor should
be the largest multiple of the radical’s index that
is less than or equal to the factor’s original
exponent.
3. Use the product property to remove factors
from the radicand.
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Example – Simplifying Radicals
Simplify:
98
98 = 49 ⋅ 2
=7 2
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Concept – Like Radicals
Two radical expressions are called like
radicals if they have the same index and
the same radicand.
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Example – Like Radicals
3 2&7 2
4 ab & − 9 ab
3
2
3
2
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Concept – Adding and Subtracting
Radical Expressions
1. Simplify each radical completely.
2. Combine like radicals by adding/subtracting
the factors in front of the radicals.
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Example – Adding and Subtracting
Radical Expressions
Subtract: 9
3−4 3
9 3−4 3 =5 3
Section 8.2 - Simplifying, Adding, & Subtracting Radical Expressions
Example – Adding and Subtracting
Radical Expressions
Add:
7 50 + 6 32
7 50 + 6 32 = 7 25 ⋅ 2 + 6 16 ⋅ 2
= 7 ⋅5 2 + 6⋅ 4 2
= 35 2 + 24 2
= 59 2
Section 8.3 – Multiplying and Dividing Radical Expressions
Concept – Multiplying Radical Expressions
If the index of each radical is the same,
multiply factors in front of the radical by
each other, and multiply the radicands by
each other. Simplify the resulting radical if
possible.
Section 8.3 – Multiplying and Dividing Radical Expressions
Example – Multiplying Radical Expressions
Multiply: 5 6 ⋅ 9 14
5 6 ⋅ 9 14 = 45 84
= 45 4 ⋅ 21
= 45 ⋅ 2 21
= 90 21
Section 8.3 – Multiplying and Dividing Radical Expressions
Concept – Multiplying Radical Expressions
If one or more of the expressions contains at
least two terms, multiply by using the
distributive property.
Section 8.3 – Multiplying and Dividing Radical Expressions
Example – Multiplying Radical Expressions
(
Multiply: 4 3 9 2 − 10 5
(
)
)
4 3 9 2 − 10 5 = 36 6 − 40 15
Section 8.3 – Multiplying and Dividing Radical Expressions
Example – Multiplying Radical Expressions
(
)(
Multiply: 7 2 + 3 3 3 2 + 4 3
(7
)(
)
)
2 + 3 3 3 2 + 4 3 = 21 4 + 28 6 + 9 6 + 12 9
= 21 ⋅ 2 + 28 6 + 9 6 + 12 ⋅ 3
= 42 + 28 6 + 9 6 + 36
= 78 + 37 6
Section 8.3 – Multiplying and Dividing Radical Expressions
Concept - Multiplying Radical Expressions
To square a radical expression containing
two or more terms, multiply the expression
by itself.
Section 8.3 – Multiplying and Dividing Radical Expressions
Example - Multiplying Radical Expressions
(
Multiply: 6 5 + 7
(6
) (
2
)(
)
2
5+ 7 = 6 5+ 7 6 5+ 7
)
= 36 25 + 6 35 + 6 35 + 49
= 36 ⋅ 5 + 6 35 + 6 35 + 7
= 180 + 6 35 + 6 35 + 7
= 187 + 12 35
Section 8.3 – Multiplying and Dividing Radical Expressions
Concept – Rationalizing a Denominator with Only
One Term Containing a Radical with an Index of n
1. Multiply both the numerator and denominator
by a radical expression that results in the
radicand being a perfect nth power.
2. Simplify the radical in the denominator.
3. Simplify the fraction by dividing out factors
that are common to the numerator and
denominator, if possible.
Section 8.3 – Multiplying and Dividing Radical Expressions
Example - Rationalizing a Denominator with Only
One Term Containing a Radical with an Index of n
4 7
Rationalize the denominator:
12
4 7 3
4 21
⋅
=
12 3
36
4 21
=
6
2 21
=
3