Download Simplifying Radical Expressions

Simplifying Radical
Expressions
Objective
I will simplify square roots and cube roots.
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.
Radical Expressions
The above symbol represents the positive or principal
root of a number.

The symbol represents the negative root of a number.
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
x8  x 4
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
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
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
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
2
2
25
43
27
5
64
1
3
8
3
27
3
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
17

17

16x
3
16x
8
4
x
x
16 x x 
16
3
8  2x x  2x
15 2
5 3
2x
2
Quotient Rule for Square Roots
If
a and b are real numbers and b  0, then
Examples:
16
16 4


81
81 9
45

49
45

49
2

25
95
3 5

7
7
2
2

5
25
a
a

b
b
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
Examples:
x 
11
x x 
x5 x
10
18x  9  2x  3 x
4
27

8
x
7
7y

25
4
27
x
8

7  y6 y
25
93
x

8
y
3
2
2
3 3

x4
7y
5
Examples:
3
88 
3
3
3
10

27
3
2
11
8 11 
3
10

3
27
10
3
3
23
3mn
n
27m n  3 m n n 
3
81

8
3 7
3
81

3
8
3
3
3 6
27  3

2
33 3
2
One Big Final Example
5
5
64x12 y 4 z18 
32  2x x y z z 
10
2 3 5
2x z
2
4 15 3
2
4 3
2x y z
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
EXIT TICKET
Simplify:
10 x  2 x 
HOMEWORK
Simplifying Square and Cube Roots WS (1-16)