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Section 10.1 Radical Expressions and Functions
Definitions and Notation
 A Radical Expression is any expression containing a radical.
𝒏
 The principal nth root of a is represented by √𝒂, where n is index and a is radicand.
 If n is 2, the nth root is called a square root and represented by √𝑎.
3
 If n is 3, the nth root is called a cube root and represented by √𝑎
5
9
4
 If n is 4 or more, fourth root( √𝑎), fifth root( √𝑎), ninth root( √𝑎), etc.
𝒏
nth root √𝒂
 If n is odd, then a can be any real number; positive and negative real numbers
 If n is even, then a must be nonnegative (0 and positive real numbers).
 For any real number a,
𝑛
1. If n is odd, √𝑎𝑛 = 𝑎
𝒏
2. If n is even, √𝒂𝒏 = |𝒂| = 𝒂
𝒏
𝒎
 Rational Exponent: √𝒂𝒎 = 𝒂 𝒏
Square Root
 If 𝒃𝟐 = 𝒂, then b is a square root of a.
 If a is a nonnegative real number and the nonnegative number b such that 𝑏 2 = 𝑎, then
b is the principal square root of a.
 A Square Root of a negative number is not a real number
 The Product and Quotient of Square Root
If a and b are real numbers, then
 √𝑎𝑏 = √𝑎 ∙ √𝑏
𝑎 √𝑎
 � =
,
𝑏≠0
𝑏 √𝑏
 Notice that the Principal Square Root NEVER BE NEGATIVE.
Square Root Functions
 A square root function is defined as
𝑓(𝑥) = √𝑥 , where 𝑥 ≥ 0.
Exercises
(Solution 1)
By Definition
The principal square root of 9 is 3,
because 32 = 9
Thus, √9 = 3
𝑚
𝑛
By Rational Exponent √𝑎𝑚 = 𝑎 𝑛
√9 ⟹ 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 2
2
√9 = �32 = 32 = 31 = 3
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 1
Section 10.1 Radical Expressions and Functions
(Solution 2)
By Definition
The principal square root of 100 is
10, because 102 = 100
Thus, −√100 = −10
𝑚
𝑛
By Rational Exponent √𝑎𝑚 = 𝑎 𝑛
−√100 ⟹ 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 2
2
−√100 = −�102 = −102 = −101 = −10
(Solution 3)
By Definition
No number gives whose square is
– 16. Thus, √−16 is not real
numbers.
Remember!
A Square Root of a negative number is
not a real number
(Solution 4)
By Definition
The principal square root of
is
1
1 2
1
, because � � =
8
8
64
1
64
1
1
=
64 8
ByQuotient rule and Rational Exponent
𝑚
𝑎 √𝑎
𝑛
�𝑎𝑚 = 𝑎 𝑛
� =
,
𝑏 √𝑏
Thus, �
�
�
Cheon-Sig Lee
www.coastalbend.edu/lee
1
⟹ 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 2
64
1
1
1
1
√1
=
=
= 2=
2
64 √64 √8
8
82
Page 2
Section 10.1 Radical Expressions and Functions
(Solution 5)
By Definition
The principal square root of
is
81
100
9
9 2
81
, because � � =
10
10
100
81
9
=−
100
10
By Quotient rule and Rational Exponent
𝑚
𝑎 √𝑎
𝑛
�𝑎𝑚 = 𝑎 𝑛
� =
,
𝑏 √𝑏
Thus, −�
−�
81
⟹ 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 2
100
81
9
√92
√81
−�
=−
=−
=−
2
100
10
√100
√10
(Solution 6)
To find 𝑓(21), substitute 21 for x
𝑓(𝑥) = √𝑥 − 5
𝑓(21) = √21 − 5 = √16 = 4
To find 𝑓(6), substitute 6 for x
𝑓(𝑥) = √𝑥 − 5
𝑓(6) = √6 − 5 = √1 = 1
To find 𝑓(5), substitute 5 for x
𝑓(𝑥) = √𝑥 − 5
𝑓(5) = √5 − 5 = √0 = 0
To find 𝑓(2), substitute 2 for x
𝑓(𝑥) = √𝑥 − 5
𝑓(2) = √2 − 5 = √−3 =?
Remember!
A Square Root of a negative number is
not a real number
Thus, 𝑓(2) = √−3 is not real number
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 3
Section 10.1 Radical Expressions and Functions
(Solution 7)
By Definition
The cube root of 216 is 6, because 63 = 216
3
Therefore, √216 = 6
𝑚
𝑛
By Rational Exponent √𝑎𝑚 = 𝑎 𝑛
3
√216 ⟹ 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 3
3
3
3
√216 = �63 = 63 = 61 = 6
(Solution 8)
By Definition
The cube root of
−125
−5
−5 3 −125
is
, because � � =
216
6
6
216
−125 −5
=
216
6
By Quotient rule and Rational Exponent
𝑚
𝑎 √𝑎
𝑛
�𝑎𝑚 = 𝑎 𝑛
� =
,
𝑏 √𝑏
3
Thus, �
3
�
−125
⟹ 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 3
216
3
3
3
−125 √−125 �(−5)3 (−5)3 −5
�
= 3
= 3
=
=
3
216
6
√216
√63
63
3
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 4