Transcript Chapter 15 - Sections 1-2-3

15.1 – Introduction to Radicals
Radical Expressions
Finding a root of a number is the inverse operation of raising
a number to a power.
radical sign
index
n
a
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.
15.1 – Introduction to Radicals
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
a is the positive square root of a and
 a is the negative square root of a.
Examples:
100  10
 0.81   0.9
 36  6
25 5

49 7
1 1
9  non-real #
15.1 – Introduction to Radicals
What does the following symbol represent?
The symbol represents the positive or
principal root of a number.
What is the radicand of the expression 4 5xy ?
5xy
15.1 – Introduction to Radicals
What does the following symbol represent?

The symbol represents the negative root of
a number.
What is the index of the expression
3
3
5x2 y5 ?
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
8 2
5
125
3

4
64
15.1 – Introduction to Radicals
nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
3  81
4
81  3
2  16
4
16  2
5
32  2
4
4
 2 
5
 32
15.1 – Introduction to Radicals
nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
5
1  1
4
16  Non-real number
6
1  Non-real number
3
27  3
15.1 – Introduction to Radicals
Radicals with Variables
x 
6 2
12
y 
6
5
 x
x  x
12
3 5
y
15
15
x y 
3
3
 y
 y
3
7 3
9
x y
21
 x9 y 21
 x y
3
Examples:
z8  z 4
3
8y
x  x
10
4
3
20
12
 2y
4x  2x
6
64x y
9
24
 4x y
3
3
8
7
15.2 – Simplifying Radicals
Simplifying Radicals using the Product Rule
Product Rule for Square Roots
If
a and b are real numbers, then a  b  a  b
Examples:
40 
4 10  4  10  2 10
18  9  2 
9 2  3 2
700  100  7  10 7
7 75  7 25  3  7  5 3 
35 3
15 
15
15.2 – Simplifying Radicals
Simplifying Radicals using the Quotient Rule
Quotient Rule for Square Roots
If
a and b are real numbers and b  0, then
Examples:
16 4
16


81
81 9
45

49
45

49
2

25
95
3 5

7
7
2
2

5
25
a
a

b
b
15.2 – Simplifying Radicals
Simplifying Radicals Containing Variables
Examples:
x 
11
x x 
x5 x
10
18x  9  2x  3x
4
27

8
x
7
7y

25
4
27
x
8
7y

7
25
93
x
8
2
2
3 3

4
x
7 y y
6

25
y3 7 y

5
15.2 – Simplifying Radicals
Simplifying Cube Roots
Examples:
3
88 
3
3
50 
3
10

27
3
3
3
81

8
8 11  2 3 11
50
3
10

3
27
3
81

3
8
3
10
3
27  3

2
33 3
2
15.2 – Simplifying Radicals
Examples:
3
27m n 
3 7
3 3 m3n6 n 
3mn
23
n
15.2 – Simplifying Radicals
Examples:
5
5
64x y z 
12
4 18
32  2x10 x 2 y 4 z15 z 3 
2x2 z3 5 2x2 y 4 z3
15.3 – Adding and Subtracting Radicals
Review and Examples:
5x  3x  8x
12 y  7 y  5y
6 11  9 11  15 11
7  3 7  2 7
15.3 – Adding and Subtracting Radicals
Simplifying Radicals Prior to Adding or Subtracting
27  75 
9  3  25  3  3 3  5 3  8 3
3 20  7 45  3 4  5  7 9  5  3  2 5  7  3 5 
6 5  21 5  15 5
36  48  4 3  9  6  16  3  4 3  3 
6 4 3  4 3 3  38 3
15.3 – Adding and Subtracting Radicals
Simplifying Radicals Prior to Adding or Subtracting
9 x4  36 x3  x3 
3x 2  6 x 2 x  x 2 x 
3x  6 x x  x x  3x  5 x x
2
2
10 3 81 p 6  3 24 p 6  10 3 27  3 p 6  3 8  3 p 6 
10  3 p
23
3 2p
23
3
28 p
30 p
23
3
23
3 2p
23
3