Download positive

7.1 – 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.
7.1 – Radicals
Radical Expressions
The above symbol represents the positive or principal
root of a number.

The symbol represents the negative root of a number.
7.1 – 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 #
7.1 Rdicals
– 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
3
8  2
x  x
3
4
x
x 
3
12
5
125
3

4
64
7.1 – 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
7.1 – 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
7.2 – 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
7.2 – Rational Exponents
m
n
n
Definition of a :
a
m
 a
or
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
7.2 – 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
7.2 – 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
7.3 – 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
7.3 – 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
7.3 – 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
7.3 – 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
7.3 – 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
7.3 – 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
7.4 – 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
7.4 – 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
7.4 – 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
7.4 – 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
7.4 – 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
7.4 – 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