Download Document

Radicals
Radical Expressions
Finding a root of a number is the inverse operation of raising
a number to a power.
This symbol is the radical or the radical sign
radical sign
index
n
a
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
Radicals
Radical Expressions
The above symbol represents the positive or principal
root of a number.

The symbol represents the negative root of a number.
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
5
25

7
49
 36  6
0.81  0.9
4
x
x 
8
9  non-real #
Radicals
Rdicals
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
3
8  2
x  x
3
4
x
x 
3
12
5
125
3

4
64
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
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
Rational Exponents
m
n
n
Definition of a :
a
m
or
 a
m
n
The value of the numerator represents the power of the
radicand.
The value of the denominator represents the index or root of
the expression.
Examples:
25
4
1
3
43
2
2 x  1 7
2
27
5
25
2
8
64
7
1
3
2 x  1
2
3
27
3
Rational Exponents
m
n
n
Definition of a :
a
m
or
 a
n
m
More Examples:
 1 
 
 27 
2
2
3
1
27
3
3
2
3
3
2
3
1
27
3
2
1
729
1
9
or
 1 
 
 27 
2
3
2
1
27
1
3
3
2
3
2
 27 
3
1
2
3
2
2
1
9
Rational Exponents
Definition of a
 mn
:
1
a
m
1
or
n
n
a
or
m
25
x
2
1
25
1
2
1
3
x
2
3
3
1
25
1
5
1
1
x
2
or
 x
3
 a
n
Examples:
 12
1
2
m
Rational Exponents
Use the properties of exponents to simplify each expression
4
x x
x
x
4
3
3
3
x
5
x
1
10
81x
12
5
2
x x
3
3 1
5 10
3x
2
4 5
3 3
2
x
x
x
2
1
3
12
x
1
x3
3
6 1
10 10
3x
4
9
x
5
10
x
1
2
2
x
1 8
12 12
x
9
12
x
3
4
Simplifying Rational Expressions
Product Rule for Square Roots
If a and b are real numbers, then a  b  a  b
Examples:
4 10  4 10  2 10
40 
7 75  7 25  3  7 25 3  7  5 3  35 3
8
4
x
x
16 x x 
16x 
17
3
16
16x  8  2 x x  2 x
17
3
15 2
5 3
2x
2
Simplifying Rational Expressions
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
Simplifying Rational Expressions
If
a and b are real numbers and b  0, then
15

3
35
3 5


3
3
90

2
9 10

2
a
a

b
b
5
9 25
9 2 5
 3 5

2
2
Simplifying Rational Expressions
Examples:
x 
11
x x 
x5 x
10
18x  9  2x  3x
4
27

8
x
7
7y

25
4
27
x
8

93
x
7 y y
6
25

8
y
3
2
2
3 3

4
x
7y
5
Simplifying Rational Expressions
Examples:
3
88 
3
3
2
11
8 11 
3
3
3
3
10
10

 3
27
27
23
3mn
n
27m n  3 m n n 
3 7
3
3
10
3
81
81
 3 
8
8
3
3
3 6
27  3

2
33 3
2
Simplifying Rational Expressions
One Big Final Example
5
5
64x y z 
12
4 18
32  2x10 x 2 y 4 z15 z 3 
2 3 5
2x z
2
4 3
2x y z
Adding, Subtracting, Multiplying Radical
Expressions
Review and Examples:
5x  3x  8x
12 y  7 y  5y
6 11  9 11  15 11
7  3 7  2 7
Adding, Subtracting, Multiplying Radical
Expressions
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
Adding, Subtracting, Multiplying Radical
Expressions
Simplifying Radicals Prior to Adding or Subtracting
9 x  36 x  x 
4
3
3
3x  6 x x  x x 
2
2
2
2
3
x
 5x x
3x  6 x x  x x 
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
Adding, Subtracting, Multiplying Radical
Expressions
If
a and b are real numbers, then a  b  a  b
7 7 
49  7
5 2 
10
6  3  18  9  2 
3 2
10 x  2 x  20x2  4  5x2 
2x 5
Adding, Subtracting, Multiplying Radical
Expressions
7


7 7 7 3 
7 3 
49  21 
7  21
5x


x 3 5 
5x  3 25x  x 5  3  5 x 
2
x 5  15 x

x 5


x 3 
x2  3x  5x  15 
x 2  3x  5x  15
Adding, Subtracting, Multiplying Radical
Expressions


36



3 6 
2
5x  4 

9  6 3  6 3  36  3  36 
33
5x  4


5x  4 
25x  4 5x  4 5x  16 
2
5 x  8 5 x  16