LESSON PPT

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
n
a
index
radical sign
radicand
The expression under the radical sign is
the radicand.
The index defines the root to be taken.

2. The symbol represents the negative root of a
number.
The above symbol represents the
positive or principal root of a number.

Radicals

3. Square Roots
If a is a positive number, then
a is the positive square root of a and
a
 is the negative square root of a.
A square root of any positive number has
two roots – one is positive and the other is
negative.
Radicals

4. Rdicals
Cube Roots
3
27 
A cube root of any positive number is positive.
Examples:
3
5
4
3
125
64

3
8
  2

A cube root of any negative number is negative.
3
a

3 3
x x 
3 12
x 4
x
Radicals

5. nth
Roots
An nth
root of any number a is a number whose
nth
power is a.
Examples:
2
4
81  3
4
16 
5
32
  2

4
3 81
4
2 16
 
5
2
  32

Radicals

6. nth
Roots
4
16
 
An nth
root of any number a is a number whose
nth
power is a.
Examples:
1

5
1
 
Non-real number
6
1
  Non-real number
3
27
  3

7.1 – Radicals

8. Rational Exponents to Radical Expression
The value of the numerator represents
the power of the radicand.
:
n
m
a
of
Definition
The value of the denominator represents
the index or root of the expression.
n m
a or  m
n
a
n
m
a

9. Rational Exponents to Radical Expression
Examples:
3
1
27
25
2
1
25
3
27
  7
2
1
2 
x
3
4
2
3
4
 
7 2
1
2 
x

10. Rational Exponents
More Examples:
3
2
3
2
27
1
3
2
27
1






3 2
3 2
27
1
9
1
3
3
729
1
3
2
3
2
27
1
3
2
27
1





  
 2
3
2
3
27
1
9
1
 
 2
2
3
1
or
Rational Exponents to Radical Expression

11. Write each radical expression into
exponential form.
1. 4.
2. 3.
5.

12. 6.
7.
8.
9.
10.

13. 11.
12. 7

14. 40 
Examples:
4 10
 
If and are real numbers, then a b
a b a b
  
Product Rule for Square Roots
2 10
7 75  7 25 3
  7 5 3
  35 3
Simplifying Rational Expressions

17
16x 
x
x16
16 x
x8
4

3 17
16x 

3 2
15
2
8 x
x 3 2
5
2
2 x
x

10
4

3
25
7

15. A radical expression is in its
simplest form when three conditions
are met:
1. No radicands have perfect square
factors other than 1
2. No radicand contains a fraction
3. No radicals appear in the
denominator of a fraction.

16. 16
81

Examples:
2
5
4
9
45
49

a
If and are real numbers and 0,then
b
a
a b b
b
 
Quotient Rule for Square Roots
2
25

9 5
7


3 5
7
16
81

2
25

45
49

7.3 – Simplifying Rational Expressions

17. 15
3

90
2

a
If and are real numbers and 0,then
b
a
a b b
b
 
3 5
3


3 5
3

 5
9 10
2


9 2 5
2
 

9 2 5
2
 
 3 5
7.3 – Simplifying Rational Expressions

18. 11
x 
Examples:
7
7
25
y

8
27
x

6
7
25
y y


3
7
5
y y
10
x x
 
5
x x
4
18x  4
9 2x
  2
3 2
x
8
9 3
x

 4
3 3
x
8
27
x

7.3 – Simplifying Rational Expressions

19. 3
88 
Examples:
3
81
8

3
10
27

3
3
81
8

3
27 3
2


3
8 11
  3
2 11
3
10
3
3
3
10
27

3
3 3
2
7.3 – Simplifying Rational Expressions
3 3 7
27m n  3 3 6
3 m n n  2 3
3mn n

20. One Big Final Example
12 4 18
5
64x y z 
10 2 4 15 3
5
32 2x x y z z
 
2 3 2 4 3
5
2 2
x z x y z
7.3 – Simplifying Rational Expressions

21. 5 3
x x
 
Review and Examples:
6 11 9 11
 
8x
15 11
12 7
y y
  5y
7 3 7
  2 7

7.4 – Adding, Subtracting, Multiplying Radical
Expressions

22. 27 75
 
Simplifying Radicals Prior to Adding or Subtracting
3 20 7 45
 
9 3 25 3
   
3 4 5 7 9 5
   
3 3 5 3
  8 3
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

23. 4 3 3
9 36
x x x
  
Simplifying Radicals Prior to Adding or Subtracting
6 6
3 3
10 81 24
p p
 
2 2 2
3 6
x x x x x
  
2
3 6
x x x x x
  
2
3 5
x x x

6 6
3 3
10 27 3 8 3
p p
   
2 2
3 3
10 3 3 2 3
p p
  
2 3
28 3
p
2 2
3 3
30 3 2 3
p p
 
7.4 – Adding, Subtracting, Multiplying Radical
Expressions

24. 5 2
 
7 7
 
10 2
x x
 
If and are real numbers, then a b
a b a b
  
10
49  7
6 3
  18  9 2
  3 2
2
20x  2
4 5x
  2 5
x
7.4 – Adding, Subtracting, Multiplying Radical
Expressions

25.  
7 7 3
  7 7 7 3
    49 21
 
 
5 3 5
x x  
  
5 3
x x
  
7 21

2
5 3 25
x x
  5 3 5
x x
  
5 15
x x

2
3 5 15
x x x
   
2
3 5 15
x x x
  
7.4 – Adding, Subtracting, Multiplying Radical
Expressions

26.   
3 6 3 6
  
 
2
5 4
x  
9 6 3 6 3 36
    3 36
 
33

  
5 4 5 4
x x
  
2
25 4 5 4 5 16
x x x
   
5 8 5 16
x x
 
7.4 – Adding, Subtracting, Multiplying Radical
Expressions