Transcript Intermediate Algebra - Seminole State College

Intermediate Algebra
Exam 4 Material
Radicals, Rational Exponents &
Equations
Square Roots
• A square root of a real number “a” is a real number that
multiplies by itself to give “a”
What is a square root of 9? 3
What is another square root of 9?  3
• What is the square root of -4 ?
Square root of – 4 does not exist in the real number system
• Why is it that square roots of negative numbers do not
exist in the real number system?
No real number multiplied by itself can give a negative answer
• Every positive real number “a” has two square roots
that have equal absolute values, but opposite signs
The two square roots of 16 are:
The two square roots of 5 are:
16 and  16 simplified: 4 and  4
5 and  5
(PositiveSquare Root : PRINCIPLEROOT )
Even Roots (2,4,6,…)
• The even “nth” root of a real number “a” is a real number
that multiplies by itself “n” times to give “a”
• Even roots of negative numbers do not exist in the real
number system, because no real number multiplied by
itself an even number of times can give a negative
number 4 16 does not exist
• Every positive real number “a” has two even roots that
have equal absolute values, but opposite signs
The fourth roots of 16: 4 16 and  4 16 simplified: 2 and  2
The fourth roots of 7: 4 7 and - 4 7
(PositiveEven Root : PRINCIPLEROOT )
Radical Expressions
• On the previous slides we have used
symbols of the form: n a
• This is called a radical expression and the
parts of the expression are named:
n
Index:
Radical Sign :
Radicand: a
• Example:
5
8
Index : 5
Radicand : 8
Cube Roots
• The cube root of a real number “a” is a
real number that multiplies by itself 3 times
to give “a”
• Every real number “a” has exactly one
cube root that is positive when “a” is
positive, and negative when “a” is negative
Only cube root of – 8: 3  8   2
3
6
Only cube root of 6:
No such thingas a principlecube root!
Odd Roots (3,5,7,…)
• The odd nth root of a real number “a” is a
real number that multiplies by itself “n”
times to give “a”
• Every real number “a” has exactly one
odd root that is positive when “a” is
positive, and negative when “a” is negative
The only fifth root of - 32: 3  32   2
The only fifth root of -7:
5 7
Rational, Irrational, and Non-real
Radical Expressions
•
n
a
is non-real only if the radicand is negative and the
index is even
6
 20 is non- real because radicandis negativeand index is even
• n a represents a rational number only if the radicand
can be written as a “perfect nth” power of an integer or
the ratio of two integers
5
•
n
a
 32 is rationalbecause  32   2
5
5
 32  2
represents an irrational number only if it is a real
number and the radicand can not be written as “perfect
nth” power of an integer or the ratio of two integers
.
4
8 is irrationalbecause 8 is not thefourthpower
4
8
of an integeror theratioof two integers
Homework Problems
• Section: 10.1
• Page: 666
• Problems: All: 1 – 6, Odd: 7 – 31,
39 – 57, 65 – 91
• MyMathLab Homework Assignment 10.1
for practice
• MyMathLab Quiz 10.1 for grade
Exponential Expressions
an
“a” is called the base
“n” is called the exponent
• If “n” is a natural number then “an” means that “a” is to be multiplied
by itself “n” times.
Example: What is the value of 24 ?
(2)(2)(2)(2) = 16
• An exponent applies only to the base (what it touches)
Example: What is the value of: - 34 ?
- (3)(3)(3)(3) = - 81
Example: What is the value of: (- 3)4 ?
(- 3)(- 3)(- 3)(- 3) = 81
• Meanings of exponents that are not natural numbers will be
discussed in this unit.
Negative Exponents: a-n
• A negative exponent has the meaning:
“reciprocate the base and make the exponent
positive”
n
1
n
a  
a
Examples:
2
1 1
3    
 3 9
2
.
3
3
 2
 3  27
    
8
 3
2
Quotient Rule for Exponential
Expressions
• When exponential expressions with the same base are divided, the
result is an exponential expression with the same base and an
exponent equal to the numerator exponent minus the denominator
exponent
am
mn

a
an
Examples:
54
4 7
3

5

5
57
.
x12
12 4
8

x

x
x4
Rational Exponents (a1/n)
and Roots
1
n
• An exponent of the form
has the meaning: “the nth root of the base, if it
exists, and, if there are two nth roots, it means
the principle (positive) one”
1
n
th
a , if it exists,is then rootof a
1
n
(If thereare two n th roots, a is theprinciple(positive)one)
1
n
( a multipliesby itself n timesto give a)
Examples of
Rational Exponent of the Form:
1/n
1
1002  10 (positivesquare root of 100)
1
2
5  5 (positivesquare rootof 5)
1
 32  (Doesnot exist!)
1
2
 3   3 (negativesquare root of 3)
1
4
7 
 9
1
7
 8
1
6

4
7
7 (positivefourthrootof 7)
 9 (seventhroot of negative9)
 (Doesnot exist!)
.
Summary Comments about
Meaning of a1/n
• When n is odd:
– a1/n always exists and is either positive,
negative or zero depending on whether “a” is
positive, negative or zero
• When n is even:
– a1/n never exists when “a” is negative
– a1/n always exists and is positive or zero
depending on whether “a” is positive or zero
Rational Exponents of the Form:
m/n
• An exponent of the form m/n has two equivalent
meanings:
(1)
am/n means find the nth root of “a”, then raise
it to the power of “m”
(assuming that the nth root of “a” exists)
(2)
am/n means raise “a” to the power of “m”
then take the nth root of am
(assuming that the nth root of “am” exists)
Example of Rational Exponent of
the Form: m/n
82/3
by definition number 1 this means find the cube root
of 8, then square it:
82/3 = 4
(cube root of 8 is 2, and 2 squared is 4)
by definition number 2 this means raise 8 to the
power of 2 and then cube root that answer:
82/3 = 4
(8 squared is 64, and the cube root of 64 is 4)
Definitions and Rules for
Exponents
• All the rules learned for natural number exponents
continue to be true for both positive and negative rational
exponents:
4
2
6
Product Rule:
aman = am+n
37  37  37
Quotient Rule:
am/an = am-n
3
3
Negative Exponents:
a-n = (1/a)n
4
7
4

7
3
.
2
7
 3

2
7
1
  
3
4
7
Definitions and Rules for
Exponents
Power Rules:
2
7
8
 74 
 3   3 49
 
 
(am)n = amn
(ab)m = ambm
(a/b)m
Zero Exponent:
.
=
am /
bm
a0 = 1 (a not zero)
2
7
3x 
2
7
 3 x
2
7
2
7
3
3
   2
4
47
0
 3
   1
 4
2
7
“Slide Rule” for Exponential
Expressions
• When both the numerator and denominator of
a fraction are factored then any factor may
slide from the top to bottom, or vice versa, by
changing the sign on the exponent
Example: Use rule to slide all factors to other
part of the fraction:
a mb  n
cr d s
 m n
r s
c d
a b
• This rule applies to all types of exponents
• Often used to make all exponents positive
Simplifying Products and
Quotients Having Factors with
Rational Exponents
• All factors containing a common base can be
combined using rules of exponents in such a way
that all exponents are positive:
• Use rules of exponents to get rid of parentheses
• Simplify top and bottom separately by using product
rules
• Use slide rule to move all factors containing a common
base to the same part of the fraction
• If any exponents are negative make a final application of
the slide rule
Simplify the Expression:


8 y y 


1
3
3
4
1
2
8y
1
6

2 y y
8 y y 2

3
4

2
3

8y y
9
12
8
3
2 1 y
16
7
12
y
2

3
2 1 y y

21 y y
1
6
6
3

2
12
21  8
7
12
y y
8
3
16
7
12
y y
32
12
39
12
Applying Rules of Exponents
in Multiplying and Factoring
• Multiply:
x
1 1
 
2 2
x
1 1
1
1
1
1
1




 
  12
 12
 x  2  x  x 2   x 2 x 2  x 2 x 2  2 x 2  2 x 2






1 1
 
2 2
1
2
 2x  2x

1
2
1
2
 x 0  x 1  2 x  2 x
1
2
 1  x 1  2 x  2 x
• Factor out the indicated factor:
x

3
4
__ __
x

3
4
4


4
5  x 




x

3
4
5x
5  x


3
4

1
2
1
2
1
4
x ;x

3
4
Radical Notation
• Roots of real numbers may be indicated
by means of either rational exponent
notation or radical notation:
n
a is calleda RADICAL(expression)
is calleda RADICALSIGN
n is calledthe INDEX
a is called the RADICAND
Notes About Radical Notation
•
•
•
•
•
If no index is shown it is assumed to be 2
When index is 2, the radical is called a “square root”
When index is 3, the radical is called a “cube root”
When index is n, the radical is called an “nth root”
In the real number system, we can only find even
roots of non-negative radicands. There are always
two roots when the index is even, but a radical with an
even index always means the positive (principle) root
• We can always find an odd root of any real number
and the result is positive or negative depending on
whether the radicand is positive or negative
Converting Between Radical and
Rational Exponent Notation
•
1.
An exponential expression with exponent of the form
“m/n” can be converted to radical notation with index of
“n”, and vice versa, by either of the following formulas:
m
n
n
m
n
 a
a  a
2
3
8 
m
m
2
3
3
 3 64  4
82
 8
2
 2  4
2.
a 
•
These definitions assume that the nth root of “a” exists
n
8 
3
2
Examples
4
7
5 
5
 5
8  8
9
3
11
4
7
OR
7
4
5
9
5
4x  4 x
11
3
OR 4
 x
11
3
.
n
x
n
.
• If “n” is even, then this notation means
principle (positive) root:
x  x
n
n
(absolutevalue needed toinsure positiveanswer)
• If “n” is odd, then:
n
x x
n
• If we assume that “x” is positive (which
we often do) then we can say that:
n
x x
n
.
Homework Problems
• Section: 10.2
• Page: 675
• Problems: All: 1 – 10, Odd: 11 – 47,
51 – 97
• MyMathLab Homework Assignment 10.2
for practice
• MyMathLab Quiz 10.2 for grade
Product Rule for Radicals
• When two radicals are multiplied that have
the same index they may be combined as
a single radical having that index and
radicand equal to the product of the two
radicands:
n
a b  ab
4
ab  a b
3
n
n
3 5  3  5  15
4
4
4
• This rule works both directions:
n
n
n
16  8 2  2 2
3
3
3
Quotient Rule for Radicals
• When two radicals are divided that have the
same index they may be combined as a single
radical having that index and radicand equal to
the quotient of the two radicands
n
n
4
a n a

b
b
4
5

7
4
5
7
• This rule works both directions:
n
.
a na
n
b
b
3
5 35
3 
8
8
3
5
2
Root of a Root Rule for Radicals
• When you take the mth root of the nth root of
a radicand “a”, it is the same as taking a
single root of “a” using an index of “mn”
.
m n
a 
4 3
6 
mn
a
12
6
NO Similar Rules for Sum and
Difference of Radicals
n
a  n b  n a b
27  8  35
3  2  3 35
3
n
.
3
3
a  n b  n a b
3
27  3 8  3 19
3  2  19
3
Simplifying Radicals
•
1.
2.
3.
4.
A radical must be simplified if any of the
following conditions exist:
Some factor of the radicand has an
exponent that is bigger than or equal to the
index
There is a radical in a denominator
(denominator needs to be “rationalized”)
The radicand is a fraction
All of the factors of the radicand have
exponents that share a common factor with
the index
Simplifying when Radicand has
Exponent Too Big
3
4
2
1. Use the product rule to write the single
radical as a product of two radicals
where the first radicand contains all
factors whose exponents match the
index and the second radicand contains
all other factors
3
33
2
2
2. Simplify the first radical
3
2 2
Example
Problem?
3
3
3
2
5
Is thereanotherexponent hat
t is toobig?
3
2
5
Writethisas a product of two radicals:
3
33
24 x y
2 3x y
2 y
3
2
3x y
2
2 y 3x y
2
2
Simplify the first radical:
Simplifying when a Denominator
Contains a Single Radical of
Index “n”
1.
2.
Simplify the top and bottom separately to get rid of
exponents under the radical that are too big
Multiply the whole fraction by a special kind of “1”
where 1 is in the form of: n m
n
m
and m is theproduct of all thefactorsrequired to
makeeveryexponentin theradicandbe equal to"n"
3.
Simplify to eliminate the radical in the denominator
Example
3
5
4 x3 y 6

3

5
22 x 3 y 6
3
y5 22 x3 y

3

5
23 x 2 y 4
5
23 x 2 y 4
5
y 5 5 22 x 3 y


35 23 x 2 y 4
y 5 25 x 5 y 5
3
y 5 22 x 3 y
35 23 x 2 y 4

2 xy 2
5
2
3 8x y

2
2 xy
4
Simplifying when Radicand is
a Fraction
1. Use the quotient rule to write the single
radical as a quotient of two radicals
2. Use the rules already learned for
simplifying when there is a radical in a
denominator
Example
5
3
4
5
3
5
4

5
5
3
2
2

5
5
3
2
2

5
23
5
3
2

5
3  23
5
5
25
24

2
Simplifying when All Exponents
in Radicand Share a Common
Factor with Index
1. Divide out the common factor from the index
and all exponents
6
4 6
8
23 x y
2
All exponentsin radicandand index share what factor?
Dividingall exponentsin and index by 2 gives :
3
23 x y  3 x
2 3
4
Problem?
3
3
33
2
2 xy
 3x3 4xy
2
Simplifying Expressions Involving
Products and/or Quotients of
Radicals with the Same Index
• Use the product and quotient rules to
combine everything under a single
radical
• Simplify the single radical by procedures
previously discussed
Example
4
ab 3 4 ab
4

a 3b 3
4
4
3
ab
a4
2 4
4
4
b
b
b
4

4
4
a
a
a
ab

a 3b 3
4
4
3
ab

a
4
a3
4
a3
Right Triangle
• A “right triangle” is a triangle that has a 900
angle (where two sides intersect
hypotenuse
perpendicularly)
c
b
900
a
• The side opposite the right angle is called
the “hypotenuse” and is traditionally
identified as side “c”
• The other two sides are called “legs” and
are traditionally labeled “a” and “b”
Pythagorean Theorem
• In a right triangle, the square of the
hypotenuse is always equal to the sum of
the squares of the legs:
c  a b
2
b
c
900
a
2
2
Pythagorean Theorem Example
• It is a known fact that a triangle having
shorter sides of lengths 3 and 4, and a
longer side of length 5, is a right triangle
with hypotenuse 5.
5
3 0
90
4
• Note that Pythagorean Theorem is true:
2
2
2
c  a b
2
2
2
5  4 3
25  16  9
Using the Pythagorean Theorem
• We can use the Pythagorean Theorem to
find the third side of a right triangle, when
the other two sides are known, by finding,
or estimating, the square root of a number
Using the Pythagorean Theorem
• Given two sides of a right triangle with one
side unknown:
– Plug two known values and one unknown
value into Pythagorean Theorem
– Use addition or subtraction to isolate the
“variable squared”
– Square root both sides to find the desired
answer
Example
• Given a right triangle with a  7 and c  25
find the other side.
c  a b
2
2
2
25  7  b
2
625  49  b
2
625 49  49  49  b
2
576  b
24  576  b
2
2
2
Homework Problems
• Section: 10.3
• Page: 685
• Problems: Odd: 7 – 19, 23 – 57,
61 – 107
• MyMathLab Homework Assignment 10.3
for practice
• MyMathLab Quiz 10.3 for grade
Adding and Subtracting Radicals
• Addition and subtraction of radicals can always be
indicated, but can be simplified into a single radical
only when the radicals are “like radicals”
• “Like Radicals” are radicals that have exactly the same
index and radicand, but may have different coefficients
Which are like radicals?
3
4
4
3 5, 4 5, - 2 5 and 3 5
• When “like radicals” are added or subtracted, the
result is a “like radical” with coefficient equal to the
sum or difference of the coefficients
34 5  24 5  54 5
-2 4 5  3 3 5 
Okay as is - can't combineunlike radicals
Note Concerning Adding and
Subtracting Radicals
• When addition or subtraction of
radicals is indicated you must first
simplify all radicals because some
radicals that do not appear to be like
radicals become like radicals when
simplified
Example
Not like terms(yet)
3
128  5 3 2  2 3 16
 3 2323
3
Simplify individual radicals:
 3 27  5 3 2  2 3 2 4
2  5 3 2  2 3 23
All like radicals :
 43 2  5 3 2  4 3 2
3
2
3 2
3
 2  23 2  5 3 2  2  2 3 2
Homework Problems
• Section: 10.4
• Page: 691
• Problems: Odd: 5 – 57
• MyMathLab Homework Assignment 10.4
for practice
• MyMathLab Quiz 10.4 for grade
Simplifying when there is a Single
Radical Term in a Denominator
1. Simplify the radical in the denominator
2. If the denominator still contains a radical,
multiply the fraction by “1” where “1” is in the
form of a “special radical” over itself
3. The “special radical” is one that contains the
factors necessary to make the denominator
radical factors have exponents equal to index
4. Simplify radical in denominator to eliminate it
Example
3
3
2
9x
3
Simplify denominator :
2
1
2
3
3 x
3
3
3
2
32 x
3
2  3x 2
3
3
3
3 x
3
3x
Multiplyby special"1":
2
3x 2
Use product rule :
Simplify denominator :
3
6x
3x
2
Simplifying to Get Rid of a Binomial
Denominator that Contains One or
Two Square Root Radicals
1. Simplify the radical(s) in the denominator
2. If the denominator still contains a radical,
multiply the fraction by “1” where “1” is in the
form of a “special binomial radical” over
itself
3. The “special binomial radical” is the conjugate
of the denominator (same terms – opposite
sign)
4. Complete multiplication (the denominator will
contain no radical)
Example
5
3 2
Radical in denominator doesn't needsimplifying
Multiplyfractionby specialone:
5
3 2

3 2 3 2
FOIL on bottom :
15  10
9 4
Simplify bottom:
15  10
3 2
Distributeon top:
15  10
Homework Problems
• Section: 10.5
• Page: 700
• Problems: Odd: 7 – 105
• MyMathLab Homework Assignment 10.5
for practice
• MyMathLab Quiz 10.5 for grade
Radical Equations
• An equation is called a radical equation if it
contains a variable in a radicand
• Examples:
x  x 3  5
x  x  5 1
3
x  4  3 2x  0
Solving Radical Equations
1. Isolate ONE radical on one side of the equal
sign
2. Raise both sides of equation to power
necessary to eliminate the isolated radical
3. Solve the resulting equation to find “apparent
solutions”
4. Apparent solutions will be actual solutions if
both sides of equation were raised to an odd
power, BUT if both sides of equation were
raised to an even power, apparent solutions
MUST be checked to see if they are actual
solutions
Why Check When Both Sides are
Raised to an Even Power?
• Raising both sides of an equation to a power does not always result
in equivalent equations
• If both sides of equation are raised to an odd power, then resulting
equations are equivalent
• If both sides of equation are raised to an even power, then resulting
equations are not equivalent (“extraneous solutions” may be
introduced)
• Raising both sides to an even power, may make a false statement
true:
2
2
4
4
 2  2 , however: - 2  2 , - 2  2 , etc.
• Raising both sides to an odd power never makes a false statement
true:
3
3
5
5
 2  2 , and: - 2  2 , - 2  2 , etc.
.
Example of Solving
Radical Equation
Check x  4
x  x 3  5
x 5  x 3
 x  5
2


x 3

2
x  10x  25  x  3
x 2  11x  28  0
x  4x  7  0
x  4  0 OR x  7  0
x  4 OR x  7
2
4 43  5 ?
4 1  5?
35
x  4 is NOT a solution
Check x  7
7  7 3  5 ?
7 4 5?
55
x7
IS a solution
Example of Solving
Radical Equation

x  x  5 1
x  5  1 x
2
x  5  1 x
 
Check x  4

2
x  5  1 2 x  x
4  2 x
2  x
2
2
 2  x
 
4x
4  4  5 1?
4  9 1?
2  3 1?
5 1
x  4 is NOT a solution
Equationhas No Solution!

Example of Solving
Radical Equation
3
x  4  3 2x  0
3

3
x  4  2x
3
x4
   2x 
3
3
3
x  4  2x
4x
(No need to check)
Homework Problems
• Section: 10.6
• Page: 709
• Problems: Odd: 7 – 57
• MyMathLab Homework Assignment 10.6
for practice
• MyMathLab Quiz 10.6 for grade