Review Homework
Pages 409-410 #1-5,12-14,16-23 (all
problems)
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
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
3
10
10

 3
27
27
10
3
23
3mn
n
27m n  3 m n n 
3 7
81

8
3
81

3
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
Real
nth Roots
Let n be an integer greater than a and let a be a real number.
If n is odd, then a has one real root:
n
a a
1
n
possible
real  n a   a
If n is even and a>0, then a has two rea
l nth roots:
nth roots
1
If n is even and a = 0, then a has one nth root:
n
0  0n  0
If n is even and a < 0, then a has no real nth roots.
1
n
nth Roots
If bn = a, then b is the nth root of a.
Roots can be written as powers 1
5
5
1
10

10
n
n
a a
Raising a root to it’s index cancels
the root and exponent leaving the
3
3
radicand. n
a a
n
( a)  a
 
n
a
n is the index, a is the radicand
n is an positive integer, a is a real number
if n is an even integer
if n is an odd integer
a<0 has no real nth roots
a<0 has one real nth root
n
a=0 has one real nth root
n
0 0
a>0 has possible two real nth roots
 a  a
n
1
n
a=0 has one real nth root
n
a a
0 0
a>0 has one real nth roots
n
1
n
a a
1
n
if n (index) is an even integer
if n is an odd integer
a<0 has no real nth roots
a<0 has one real nth root
2 16  4 i (not a real solution)
a=0 has one real nth root
40  0
3 8  2
a=0 has one real nth root
30  0
a>0 has two possible real nth roots a>0 has one real nth roots
4
x
 32
4 32  2 4 2
3 27  3
th
(n
Roots with variables
Section 6-4
n
x
roots)
n is the index, x is the radicand
x x
2
3
2 4
3
x x
3
3
2 8
4
x x
2
4
x
4
3
4
6
9
8
12
x
3
6
(2)  64  4
6
3
9
3
3
(-2)   512  8
9
3
2 4
4
(2)  256  4
2 8
4
(2)  4096  8
8
12
but
so !!!!
4
8
12
4
4
x  x and (-2)  8
12
4
3
3
x | x |
12
3
Remember
• Even, Even, Odd
When the index is even and the radicand has even
exponent, and if the solution includes a variable
to an odd power, the absolute value of the
variable must be used.
4
12
x
 x
x x
8
odd
4
3
(even, even, odd)
(even, even, even - not necessary)
never use | |
Practice Worksheet
Operations with Radical Expressions
Homework
Finish worksheet
page 409-410
#6, 15, 24-37, 47-54
Older edition (pages 433-434)