15.1 – Introduction to Radicals Radical Expressions Finding a root of a number is the inverse operation of raising a number to a.
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Transcript 15.1 – Introduction to Radicals Radical Expressions Finding a root of a number is the inverse operation of raising a number to a.
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
2
5
5x y ?
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
y
15
y
3
x y
3
3
7 3
9
x y
21
x9 y 21
x y
3
Examples:
4
z
z
8
3
x
8 y12 2y 4
20
x
10
3
4x 2x
6
3
64x 9 y 24 4x3 y8
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
a
a and b are real numbers and b 0, then
b
b
Examples:
16
16 4
81
81 9
45
49
45
49
2
25
95 3 5
7
7
2
2
5
25
15.2 – Simplifying Radicals
Simplifying Radicals Containing Variables
Examples:
x
11
x x
10
x5 x
18x 9 2x 3x
4
27
8
x
7
7y
25
4
27
x
8
7y
7
25
93
x8
2
2
3 3
x4
7 y6 y
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 11
3
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
33 m3n6n
3mn
23
n
15.2 – Simplifying Radicals
Examples:
5
5
12
4 18
64x y z
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:
5 x 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 38 3
15.3 – Adding and Subtracting Radicals
Simplifying Radicals Prior to Adding or Subtracting
9x4 36x3 x3
3x2 6 x2 x x2 x
3x 6x x x x 3x 5x x
2
2
10 3 81 p 6 3 24 p 6
10 3 p
23
3 2p
23
10 3 27 3 p 6 3 8 3 p 6
3
28 p
30 p
23
3
23
3 2p
23
3