Changing Bases
Base 10: example number 2120
10³
10²
10¹
2
1
2
10⁰
Implied base 10
0 ₁₀
10³∙2 + 10²∙1 + 10¹∙2 + 10⁰∙0 = 2120₁₀
Base 8: 4110₈
8³
8²
8¹
8⁰
4
1
1
0₈
Base 8
8³∙4 + 8²∙1 + 8¹∙1 + 8⁰∙0 = 2120₁₀
Problem Solving:
3, 2, 1, … lets go!
Express the base 4 number 321₄ as
a base ten number.
Answer:
57
Add:
23₄ + 54₈ = _______₁₀
(Base 10 number)
Answer:
55
Subtract:
123.11₄ - 15.23₆ =
______₁₀
(Base 10 number)
Answer:
15 ⁴³⁄₄₈
Express the base 10 number 493 as
a base two number.
Answer:
111101101₂
Add:
347.213₁₀ + 11.428₁₀ =
________₁₀
(Base 10 number)
Answer:
358.641
Factorials
Factorial symbol ! is a shorthand
notation for a special type of
multiplication.
N! is written as
N∙(N-1)∙(N-2)∙(N-3)∙ ….. ∙1
Note: 0! = 1
Example: 5! = 5∙4∙3∙2∙1
= 120
Problem Solving:
3, 2, 1, … lets go!
Solve:
6! = _____
Answer:
720
Solve:
5!
3!
Answer:
20
Solve:
5!
3!2!
Answer:
10
Squares
Positive Exponents
“Squared”: a² = a·a
example: 3² = 3·3
=9
0²=0
1²=1
2²=4
3²=9
4²=16
5²=25
6²=36
7²=49
8²=64
9²=81
10²=100
11²=121
12²=144
13²=169
15²=225
16²=256
20²=400
25²=625
What is the sum of the
first 9 perfect squares?
Answer:
1+4+9+16+25+36+49+64+81=
285
Shortcut:
Use this formula
n(n+1)(2n+1)
6
Shortcut:
Use this formula
9(9+1)(2∙9+1)
6
Answer: 285
Square Roots
Review
9.1
Evaluating Roots
1. Find square roots.
2. Decide whether a given root is rational,
irrational, or not a real number.
3. Find decimal approximations for
irrational square roots.
4. Use the Pythagorean formula.
5. Use the distance formula.
6. Find cube, fourth, and other roots.
9.1.1: Find square roots.
• When squaring a number, multiply the number by itself. To
find the square root of a number, find a number that when
multiplied by itself, results in the given number. The number a is
called a square root of the number a 2.
Find square roots. (cont’d)
The positive or principal square root of a number is written with
the symbol
The symbol –
The symbol
.
is used for the negative square root of a number.
, is called a radical sign, always represents the
positive square root (except that 0  0 ). The number inside the
radical sign is called the radicand, and the entire expression—radical
sign and radicand—is called a radical.
Radical Sign
Radicand
a
Find square roots. (cont’d)
The statement 9  3 is incorrect. It says, in part, that a positive
number equals a negative number.
EXAMPLE 1
• Find all square roots of 64.
Solution:
Positive Square Root
64  8
Negative Square Root
 64  8
Finding All Square
Roots of a
Number
EXAMPLE 2:
Finding Square Roots
•Find each square root.
Solution:
169
 225
25
64
 13
 15

25
64
5

8
EXAMPLE 3:
Squaring Radical Expressions
•Find the square of each radical expression.
Solution:

17
 29
2x  3
2
 17 

2
  29


 17

2
2x  3
2
 29

2
 2x2  3
9.1.2: Deciding whether a given root is rational,
irrational, or not a real number.
All numbers with square roots that are rational are called
perfect squares.
Perfect Squares
Rational Square Roots
25
25  5
144
144  12
4
9
4 2

9 3
A number that is not a perfect square has a square root that is
irrational. Many square roots of integers are irrational.
Not every number has a real number square root. The
square of a real number can never be negative. Therefore, -36
is not a real number.
EXAMPLE 4:
Identifying Types of
Square Roots
•Tell whether each square root is rational,
irrational, or not a real number.
Solution:
27
irrational
2
36  6 rational
27
not a real number
Not all irrational numbers are square roots of integers. For
example  (approx. 3.14159) is a irrational number that is not an
square root of an integer.
What is a right triangle?
hypotenuse
leg
right angle
leg
It is a triangle which has an angle that is
90 degrees.
The two sides that make up the right angle
are called legs.
The side opposite the right angle is the
hypotenuse.
The Pythagorean Theorem
In a right triangle, if a and b are the
measures of the legs and c is the
hypotenuse, then
a2 + b2 = c2.
Note: The hypotenuse, c, is always the
longest side.
Find the length of the
hypotenuse if
1. a = 12 and b2 = 16.
2
2
12 + 16 = c
144 + 256 = c2
400 = c2
Take the square root of both sides.
2
400  c
20 = c
Find the length of the hypotenuse if
2. a = 5 and b = 7.
5 2 + 7 2 = c2
25 + 49 = c2
74 = c2
Take the square root of both sides.
74  c
8.60 = c
2
Find the length of the hypotenuse
given a = 6 and b = 12
1.
2.
3.
4.
180
324
13.42
18
Find the length of the leg, to the
nearest hundredth, if
3. a = 4 and c = 10.
42 + b2 = 102
16 + b2 = 100
Solve for b.
16 - 16 + b2 = 100 - 16
b2 = 84
2
b  84
b = 9.17
Find the length of the leg, to the
nearest hundredth, if
4. c = 10 and b = 7.
a2 + 72 = 102
a2 + 49 = 100
Solve for a.
a2 = 100 - 49
a2 = 51
2
a  51
a = 7.14
Find the length of the missing side
given a = 4 and c = 5
1.
2.
3.
4.
1
3
6.4
9
5. The measures of three sides of a
triangle are given below. Determine
whether each triangle is a right triangle.
73 , 3, and 8
Which side is the biggest?
The square root of 73 (= 8.5)! This must be
the hypotenuse (c).
Plug your information into the Pythagorean
Theorem. It doesn’t matter which number
is a or b.
Sides: 73 , 3, and 8
32 + 82 = ( 73 ) 2
9 + 64 = 73
73 = 73
Since this is true, the triangle is a
right triangle!! If it was not true, it
would not be a right triangle.
Determine whether the triangle is a right
triangle given the sides 6, 9, and 45
1. Yes
2. No
3. Purple
EXAMPLE 6
Using the Pythagorean Formula
Find the length of the unknown side in each right triangle.
Solution:
a  7, b  24
7 2  242  c 2
49  576  c 2
c  625
a 2  132  152
c  15, b  13
11
8
?
 25
a 2  169  225
a  56
82  b 2  112
b  57
625  c 2
a 2  56
 7.483
64  b 2  121
 7.550
b 2  57
EXAMPLE 7
Using the Pythagorean Formula
to Solve an Application
A rectangle has dimensions of 5 ft by 12 ft. Find the length
of its diagonal.
12 ft
5 ft
Solution:
52  122  c 2
25  144  c
2
169  c 2
c  169
c  13ft
9.1.5: Use the distance formula.
The distance between the points
d
 x1, y1 and  x2 , y2 is
 x2  x1    y2  y1 
2
2
.
EXAMPLE 8
Using the Distance Formula
• Find the distance between  6,3 and  2, 4 .
Solution:
d
 2   6   4  3
d  42   7 
d  16  49
d  65
2
2
2
9.1.6: Find cube, fourth, and other roots.
• Finding the square root of a number is the inverse of
squaring a number. In a similar way, there are inverses to
finding the cube of a number or to finding the fourth or
greater power of a number.
• The nth root of a is written
In
n
n
a.
a , the number n is the index or order of the radical.
Index
Radical sign
n
Radicand
a
It can be helpful to complete and keep a list to refer to of third and
fourth powers from 1-10.
EXAMPLE 9
Finding Cube Roots
• Find each cube root.
•
Solution:
3
64
4
3
27
 3
3
512
EXAMPLE 10
Finding Other Roots
• Find each root.
Solution:
4
81
 4 81
3
 3
4
81
Not a real number.
5
243
3
5
243
 3
9.2
Evaluating Roots
1. Multiply square root radicals.
2. Simplify radicals by using the product rule.
3. Simplify radicals by using the quotient
rule.
4. Simplify radicals involving variables.
5. Simplify other roots.
9.2.1: Multiply square root
radicals.
• For nonnegative real numbers a and b,
a  b  a  b and
a  b  a  b.
• That is, the product of two square roots is the square root of
the product, and the square root of a product is the product
of the square roots.
It is important to note that the radicands not be negative numbers in the
product rule. Also, in general, x  y  x  y .
EXAMPLE 1
Using the Product Rule to
Multiply Radicals
•Find each product. Assume that x  0.
Solution:
3 5
 3 5
 15
6  11
 6 11
 66
13  x
 13  x
 13x
10  10
 10 10
 100
 10
9.2.2: Simplify radicals using the product rule.
• A square root radical is simplified when no
perfect square factor remains under the
radical sign.
• This can be accomplished by using the
product rule:
a b  a  b
EXAMPLE 2
Using the Product Rule to
Simplify Radicals
•Simplify each radical.
Solution:
60
 4  15
 2 15
500
 100  5
 10 5
17
It cannot be simplified further.
EXAMPLE 3
Multiplying and Simplifying
Radicals
•Find each product and simplify.
Solution:
10  50
6 2
 10  50
 500
 100  5
 62
 12
2 3
 10 5
9.2.3: Simplify radicals by using the quotient
rule.
• The quotient rule for radicals is similar to the
product
• rule.
EXAMPLE 4
Using the Quotient Rule to
Simply Radicals
•Simplify each radical.
Solution:
4
49
4

49
48
3

48
3
 16
5
36
5

36
5

6

2
7
4
EXAMPLE 5
Using the Quotient Rule to
Divide Radicals
• Simplify.
Solution:
8 50
4 5
8 50
 
4
5
50
 2
5
 2  10
 2 10
EXAMPLE 6
Using Both the Product
and Quotient Rules
• Simplify.
Solution:
3 7

8 2

3 7

8 2

21
16

21
16

21
4
9.2.4: Simplify radicals involving variables.
• Radicals can also involve variables.
• The square root of a squared number is
always nonnegative. The absolute value is used
to express this.
2
For any real number a,
a  a.
• The product and quotient rules apply when
variables appear under the radical sign, as long
as the variables represent only nonnegative
real numbers
x  0, x  x.
Simplifying Radicals Involving
Variables
EXAMPLE 7
•Simplify each radical. Assume that all
variables represent positive real numbers.
Solution:
x
6
x
3
100 p 8
 100  p8
7
y4

7
y4
Since  x
 10 p 4
7
 2
y

3 2
 x6
9.2.5: Simplify other roots.
• To simplify cube roots, look for factors that are perfect
cubes. A perfect cube is a number with a rational cube root.
• For example, 3 64  4 , and because 4 is a rational number,
64 is a perfect cube.
• For all real number for which the indicated roots exist,
n
a  n b  n ab and
n
a na

b  0 .
n
b
b
EXAMPLE 8
Simplifying Other Roots
•Simplify each radical.
Solution:
3
108
 3 27  3 4
 33 4
4
160
 4 16 10
 4 16  4 10
4
16
625
4
16
4
625
2

5
 2 4 10
Simplify other roots. (cont’d)
• Other roots of radicals involving variables
can also be simplified. To simplify cube roots
with variables, use the fact that for any real
number a,
3
a3  a.
• This is true whether a is positive or negative.
Simplifying Cube Roots Involving
Variables
EXAMPLE 9
•Simplify each radical.
Solution:
3
z
9
 z3
3
8x 6
 3 8  3 x6
3
54t 5
 3 27t 3  2t 2
15
3
a
64
3
15
a
 3
64
 2x 2
 3 27t 3  3 2t 2
a5

4
 3t 3 2t 2
9.3
Adding and Subtracting Radicals
1. Add and subtract radicals.
2. Simplify radical sums and differences.
3. Simplify more complicated radical
expressions.
9.3.1: Add and subtract radicals.
• We add or subtract radicals by using the distributive
property. For example,
8 36 3
(8  6) 3
 14 3.
Only like radicals—those which are multiples of the same
root of the same number—can be combined this way. The
preceding example shows like radicals. By contrast, examples of
unlike radicals are
2 5 and 2 3,
as well as 2 3 and 2 3 3.
Note that
5 + 3 5 cannot be simplified.
Radicands are different
Indexes are different
EXAMPLE 1
Adding and Subtracting
Like Radicals
• Add or subtract, as indicated.
8 52 5
3 11  12 11
7  10
Solution:
 8  2  5
  3  12  11
 10 5
 9 11
It cannot be
added by the
distributive
property.
9.3.2: Simplify radical sums and
differences.
• Sometimes, one or more radical expressions in
a sum or difference must be simplified. Then,
any like radicals that result can be added or
subtracted.
Adding and Subtracting Radicals
That Must Be Simplified
EXAMPLE 2
•Add or subtract, as indicated.
27  12
Solution:
3 32 3
5 3
2 3 54  4 3 2
5 200  6 18

 5
5
  
2   6 9  2 
100  2  6
100 
92
2

3

27  3 2  4 3 2


 2 3 3 2  4 3 2
 50 2 18 2
 63 2  43 2
 32 2
 10 3 2
9.3.3: Simplify more complicated radical
expressions.
• When simplifying more complicated radical
expressions, recall the rules for order of
operations.
A sum or difference of radicals can be simplified only if the
radicals are like radicals. Thus, 5  3 5  4 5, but 5  5 3
cannot be simplified further.
EXAMPLE 3A
Simplifying Radical Expressions
•Simplify each radical expression. Assume that
all variables represent nonnegative real
numbers.
7  21  2 27
6  3r  8r
Solution:
 7  21  2 27
 7 3  2 27
 147  2 27
 6  r  2 2r
 3 2r  2 2r
 7 3  2 3 3
 6  3r  2 2r
 5 2r
 49  3  2 27
 7 36 3
 18r  2 2r
 49  3  2 27
 13 3
 9  2r  2 2 r
 
EXAMPLE 3B
Simplifying Radical
Expressions (cont’d)
•Simplify each radical expression. Assume that
all variables represent nonnegative real
numbers.
2
3
3
4
4
y 72  18 y
Solution:
y
 

9 8 

 
9  2 y2
 y 3 8  3 2 y2
    3
 y 3 2 2

 
y 6 2  3

2 y2
2  y2



81x  5 24 x

 
 6 2y 3 2y

 3 2y
 3x  3 3x  5  2 x   3 3x
 3y 2
 3x  3 3x  10 x  3 3x
3
27 x3  3 3x  5

 13x 3 3x
 
3
8 x3  3 3x


9.4
Rationalizing the Denominator
1. Rationalize denominators with square
roots.
2. Write radicals in simplified form.
3. Rationalize denominators with cube
roots.
9.4.1: Rationalize denominators
with square roots.
• It is easier to work with a radical expression if the
denominators do not contain any radicals.
1
1 2
2


2
2 2 2
This process of changing the denominator from a radical, or
irrational number, to a rational number is called rationalizing
the denominator. The value of the radical expression is not
changed; only the form is changed, because the expression has
been multiplied by 1 in the form of
2
.
2
EXAMPLE 1
Rationalizing Denominators
• Rationalize each denominator.
Solution:
18
24
16
8
18 6


2 6 6
18 6

26
18 6

12
16
2


2 2 2
16 2

22
16 2

4
3 6

2
4 2
9.4.2: Write radicals in simplified
form.
A radical is considered to be in simplified form if the
following three conditions are met.
1. The radicand contains no factor (except 1) that is
a perfect square (in dealing with square roots), a
perfect cube (in dealing with cube roots), and so
on.
2. The radicand has no fractions.
3. No denominator contains a radical.
EXAMPLE 2
Simplifying a Radical
5
Simplify
.
18
Solution:
5

18
5 18


18 18
5  18

18
5  92

18
5 9 2

18
3 5  2

18
3 10

18
10

6
EXAMPLE 3
•Simplify
Simplifying a Product
of Radicals
1 5
 .
2 6
Solution:
1 5


2 6
5

12
5 3

6
15

6
5

12
5
3


2 3 3
EXAMPLE 4
Simplifying a Quotient
of Radicals
5p
•Simplify q . Assume that p and q are
positive numbers.
Solution:

5p
q

q
q
5 pq

q
EXAMPLE 5
Simplifying a Radical Quotient
5r 2t 2
7
• Simplify
. Assume that r and t
represent nonnegative real numbers.
Solution:
5r 2 t 2

7
5r 2 t 2
7


7
7
35r 2t 2

7

rt 35
7
9.4.3: Rationalize denominators
with cube roots.
EXAMPLE 6
Rationalizing Denominators with
Cube Roots
•Rationalize each denominator.
Solution:
5
6
5 3 62
3 
6 3 62
2
3
3
2 3 32
 3 
3 3 32
3
3
3
3
3
3
,x  0
4x
3


3
5  62
3
3
180

6
63
2  32
3

33
3 3 42 x 2
3

4 x 3 42 x 2
3
3
3
18
3
3 2 8x2


3 3 3
4x
4 x
3
3
6 x2
8  3 6 x2


2x
4x
3
3 16 x 2
3
9.5
More Simplifying and Operations
with Radicals
1. Simplify products of radical expressions.
2. Use conjugates to rationalize
denominators of radical expressions.
3. Write radical expressions with quotients
in lowest terms.
More Simplifying and Operations with Radicals
The conditions for which a radical is in simplest form
were listed in the previous section. A set of guidelines to
use when you are simplifying radical expressions follows:
More Simplifying and Operations with
Radicals (cont’d)
9.5.1: Simplify products of
radical expressions.
Multiplying Radical
Expressions (cont’d)
EXAMPLE 1A
• Find each product and simplify.
2

8  20



2 5 3 
32 2

Solution:

 2 2 2  45

 2 2 2


4 5
 2 2 2 2 5
 42 5 2
 4  2 10


 2

 3   2  2 2   5 3  3   5 3  2 2 
 6  11  10 6
 11  9 6
EXAMPLE 1B
Multiplying Radical
Expressions
• Find each product and simplify.


2 5 
10  2

Solution:
 2
 10   2  2   5  10   5  2 
 20  2  50  10
 2 5  2  5 2  10
Using Special Products
with Radicals
EXAMPLE 2
•Find each product. Assume that x ≥ 0.

5 3

2
4
2 5



2  x 
2
2
Solution:

 5
2
2
 5   3  3
2
 4 2
2


 2 4 2  5  5
 56 5 9
 32  40 2  25
 14  6 5
 57  40 2
2
Remember only like radicals can be combined!
 2  2  2
2
 x x
 44 x  x
2
Using a Special Product with
Radicals.
Example 3 uses the rule for the product of the sum and
difference of two terms,
2
2
x

y
x

y

x

y
.



EXAMPLE 3
Using a Special Product
with Radicals
•Find each product. Assume that y  0.

32

Solution:


 
3
 3 4
 1
32
2
  2

2

y 4


 
y 4
2
y
 y  16
  4

2
9.5.2: Use conjugates to rationalize
denominators of radical expressions.
• The results in the previous example do not
contain radicals. The pairs being multiplied are
called conjugates of each other. Conjugates
can be used to rationalize the denominators in
more complicated quotients, such as 2 .
4 3
To simplify a radical expression, with two terms in
the denominator, where at least one of the terms is a
square root radical, multiply numerator and
denominator by the conjugate of the denominator.
Using Conjugates to
Rationalize Denominators
EXAMPLE 4A
Simplify by rationalizing each denominator.
5+3
2 5
3
2 5
Solution:

3
2 5

2 5 2 5



2  5
32  5 

3 2 5
2
2
45


3 2 5



1

 3 2  5


  2  5 
5 2  5 
5 3
2
2 5 563 5
2 2  52
5 5  11

45

5 5  11
1


  5 5  11
 11  5 5
Using Conjugates to
Rationalize Denominators
(cont’d)
EXAMPLE 4B
Simplify by rationalizing each denominator.
Assume that t  0.
3
2 t
Solution:


3
2 t

2 t 2 t

2


 t 
3 2 t
2
2

3 2 t
4t

9.5.3: Write radical expressions
with quotients in lowest terms.
EXAMPLE 5
5 3  15
10
• Write
Solution:

5

3 3
10
3 3

2

Writing a Radical Quotient
in Lowest Terms
in lowest terms.
9.6
Solving Equations with Radicals
1. Solve radical equations having square root
radicals.
2. Identify equations with no solutions.
3. Solve equations by squaring a binomial.
4. Solve radical equations having cube root
radicals.
Solving Equations with Radicals.
• A radical equation is an equation having a
variable in the radicand, such as
x 1  3
or
3 x  8x  9
9.6.1: Solve radical equations having square
root radicals.
To solve radical equations having square root
radicals, we need a new property, called the squaring
property of equality.
If each side of a given equation is squared, then all
solutions of the original equation are among the
solutions of the squared equation.
Be very careful with the squaring property: Using this property can give a
new equation with more solutions than the original equation has. Because of this
possibility, checking is an essential part of the process. All proposed solutions
from the squared equation must be checked in the original equation.
Using the Squaring
Property of Equality
EXAMPLE 1
• Solve.
Solution:

9 x  4
9 x

2
4
2
9  x  16
9  x  9  16  9
x  7
x  7
7
It is important to note that even though the algebraic work may be done
perfectly, the answer produced may not make the original equation true.
EXAMPLE 2
Using the Squaring Property
with a Radical on Each Side
•Solve.
3x  9  2 x
Solution:

3x  9
  2 x 
2
2
3x  9  4x
3x  9  3x  4x  3x
x 9
9
9.6.2: Identify equations with
no solutions.
Using the Squaring Property
when One Side Is Negative
EXAMPLE 3
•Solve.
x  4
Solution:
 x
2
  4 
Check:
2
x  4
16  4
4  4
False
x  16

Because x represents the principal or nonnegative square root of x in Example 3,
we might have seen immediately that there is no solution.
Solving a Radical Equation.
• Use the following steps when solving an equation with
radicals.
• Step 1
Isolate a radical. Arrange the terms so that
a radical is isolated on one side of the
equation.
Step 2
Square both sides.
Step 3
Combine like terms.
Step 4
Repeat Steps 1-3 if there is still a term with a
radical.
Step 5
Solve the equation. Find all proposed solutions.
Step 6
Check all proposed solutions in the original
equation.
Using the Squaring Property
with a Quadratic Expression
EXAMPLE 4
•Solve
x  x2  4x 16.
Solution:
x2 

x 2  4 x  16

2
x 2  x 2  x 2  4 x  16  x 2
0  4x  4x 16  4x
4 x 16

4
4
x  4
Since x must be a positive number the
solution set is Ø.
9.6.3: Solve equations by
squaring a binomial.
EXAMPLE 5
• Solve
Using the Squaring Property
when One Side Has Two Terms
2 x  1  10 x  9.
 2 x  1   10 x  9 
4 x2  4 x  1  10 x  9  10 x  9  10 x  9
2
Solution:
2
4 x 2  14 x  8  0
 2x  1 2x  8  0
2x 1  0
or
2x  8  0
x4
1
x
2
Since x must be positive the solution set is {4}.
EXAMPLE 6
25 x  6  x
• Solve.
Solution:
Rewriting an Equation before
using the Squaring Property
25 x  6  6  x  6

25 x

2
  x  6
2
25 x  25 x  x 2  12 x  36  25 x
0  x 2  13x  36
0   x  4 x  9 
or
0  x4
0  x 9
x4
x 9
The solution set is {4,9}.
Solve equations by squaring a binomial.
• Errors often occur when both sides of an equation are squared. For
instance, when both sides of
9x  2x 1
•
are squared, the entire binomial 2x + 1 must be squared to get 4x2 + 4x +
1. It is incorrect to square the 2x and the 1 separately to get 4x2 + 1.
Using the Squaring
Property Twice
EXAMPLE 7
•Solve.
x 1  x  4  1
Solution:

x 1  1 x  4
x 1
  1 
2
x4

2
x  1  1  2 x  4   x  4

4  2 x4
2
16  4x 16
32 4x

4
4
x 8
The solution set is {8}.

2
9.6.4: Solve radical equations
having cube root radicals.
Solve radical equations having cube
root radicals.
We can extend the concept of raising both sides of
an equation to a power in order to solve radical
equations with cube roots.
Solving Equations with
Cube Root Radicals
EXAMPLE 8
•Solve each equation.
3
Solution:

3
7 x  3 4x  2
 
3
7x

3
4x  2
7 x  4x  2
3x 2

3 3
2
x
3
2
 
3
3

3
x2  3 26 x  27
 x  
3
2
3
3
26 x  27

3
x 2  26 x  27
0  x 2  26  27
0   x  27  x 1
or
0  x  27
0  x 1
x  27
x 1
27,1
Rational Exponents
Fraction Exponents
Radical expression and Exponents
By definition of Radical Expression.
5  125 so
3
3
125  5
The index of the Radical is 3.
How would we simplify this expression?
What does the fraction exponent do to the
number?
9
1
2
The number can be written as a Radical
expression, with an index of the denominator.
2
9
The Rule for Rational Exponents
1
n
b  b
n
1
3
64  64  4
3
Write in Radical form
1
6
a 
1
2
m 
Write in Radical form
1
6
a  a
1
2
6
m  m
Write each Radical using Rational
Exponents
5
b
w
Write each Radical using Rational
Exponents
5
b b
1
5
ww
1
2
What about Negative exponents
Negative exponents make inverses.
49
1
2

1
49
1
2
1

7
What if the numerator is not 1
Evaluate
2
5
32  32
5
2
What if the numerator is not 1
Evaluate
2
5
32  32
5
5
2 
5 2
2 4
2
2
 2
5
10
For any nonzero real number b, and integer
m and n
Make sure the Radical express is real, no b<0
when n is even.
m
n
b  b or
n
m
 b
n
m
Simplify
6
16
3
2
Simplify
1
6
6
16 16
 1
3
2
23

1
4 6
(2 )
2

2
2
2
3
1
3
1
3

2
2
2
3
4
6
1
3
1
3
1
3
2 2 2 3 2
Simplify
6
4x
4
Simplify
6
1
6
4x  4 x
4
 x
 2
1
2 6
2
6
4
6
4
6
4
6
1
3
2 x 2 x
 
 2x
1
2 3
2
3
 3 2x2
Competition Problems
Which number does not belong in
the set?
A.
B.
C.
D.
Which number does not belong in
the set?
A.
B.
C.
D.
Solve
Answer:
100/9
Simplify
Answer:
Solve for
Answer:
3