Chapter 15
Roots and Radicals
Section 15.1
Introduction to Radicals
Terminology
• The symbol
is called a radical or radical sign
• In the expression 𝑎, 𝑎 is called the radicand
• An expression that has a radical sign is called a radical
expression
• We can use it to mean any number of roots, e.g.
3
27 is the
cubed root of 27, or 3, 33 = 27. If there is no number in the
“vee” then it means square root
3
Square Roots
• -2 is the negative square root of 4
• 2 is the positive square root of 4
• Your book defines 4 = 2 and - 4 = -2;
However, if x2=4, x =  2; be careful in solving problems!
• If a number, a, is <0, then its square root is not a real number
4
Perfect Squares and Irrational Numbers
• a2 is a perfect square; its square root is a
• The square root of a perfect square is a rational number,
3
2
e.g., 2 = 4 , 4 = 16 , =
9
4
Remember that the definition of a rational number is a
number that can be expressed as a fraction
• Square roots of numbers that are not perfect squares are
called irrational, e.g., 2 , 11 , etc.
•
0 = 0!!!
5
Examples
Find the roots:
64𝑥 10 =
•
•
•
3
−8𝑥 6 =
25𝑎2 𝑏 20 =
• −
3
64𝑎3
𝑏9
=
6
Solutions
64𝑥 10 = 8x5
•
•
•
3
−8𝑥 6 = -2x2
25𝑎2 𝑏 20 = 5ab10
• −
3
64𝑎3
𝑏9
=
−4𝑎
𝑏3
7
Chapter 15.2
Simplifying Radicals
8
Multiplying and Dividing
• If 𝑎 and 𝑏 are real numbers, then
𝑎 𝑏 = 𝑎𝑏
We can treat the radical like parentheses
•
𝑎
𝑏
=
𝑎
𝑏
, for a and b real and b ≠ 0
Again, treat the radical like parentheses
9
Examples
•
40 = 4 10 = 2 10
•
5
16
•
𝑥 7 = 𝑥𝑥 6 = 𝑥 𝑥 6 =𝑥 3 𝑥
=
5
16
=
5
4
10
Examples
•
45 =
•
18 =
•
125 =
•
125
36
•
16
81
=
=
11
Solutions
•
45 = 3 5
•
18 = 3 2
•
125 = 5 5
•
125
36
•
16
81
=
=
5 5
36
4
9
12
Examples
•
𝑥2
5
•
45𝑥 5 =
•
6
6
=
=
13
Solutions
•
𝑥2
5
•
45𝑥 5 = 3𝑥 2 5𝑥
•
6
6
=
=
𝑥
5
1
6
14
Rationalizing Denominators
• Many people don’t like square roots in the denominators
• To make them happy, we multiply the top and bottom by the
square root of the denominator
•
𝑎
𝑏
=
𝑎 𝑏
𝑏 𝑏
𝑎𝑏
𝑏
• Unless you are asked to do this, I will accept either
15
Example: Rationalize the Denominator
•
•
2
5
=
7
11
=
Solutions
•
•
2
5
=2
7
11
=
5
5
77
11
17