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# Intermediate Algebra

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```Intermediate Algebra
Exam 4 Material
Radicals, Rational Exponents &
Equations
Square Roots
вЂў A square root of a real number вЂњaвЂќ is a real number that
multiplies by itself to give вЂњaвЂќ
What is a square root of 9? 3
What is another square root of 9? пЂ­ 3
вЂў What is the square root of -4 ?
Square root of вЂ“ 4 does not exist in the real number system
вЂў 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
вЂў Every positive real number вЂњaвЂќ 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:
(Positive
16 and пЂ­ 16
5 and пЂ­ 5
Square Root : PRINCIPLE
simplified
ROOT)
: 4 and пЂ­ 4
Even Roots (2,4,6,вЂ¦)
вЂў The even вЂњnthвЂќ root of a real number вЂњaвЂќ is a real number
that multiplies by itself вЂњnвЂќ times to give вЂњaвЂќ
вЂў 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 пЂ­ 16 does not exist
вЂў Every positive real number вЂњaвЂќ has two even roots that
have equal absolute values, but opposite signs
The fourth roots of 16: 4 16 and пЂ­ 4 16 simplified : 2 and пЂ­ 2
The fourth roots of 7: 4 7 and - 4 7
(Positive
Even Root : PRINCIPLE
ROOT)
вЂў On the previous slides we have used
symbols of the form: n a
вЂў This is called a radical expression and the
parts of the expression are named:
n
Index:
вЂў Example:
5
8
Index : 5
: 8
Cube Roots
вЂў The cube root of a real number вЂњaвЂќ is a
real number that multiplies by itself 3 times
to give вЂњaвЂќ
вЂў Every real number вЂњaвЂќ has exactly one
cube root that is positive when вЂњaвЂќ is
positive, and negative when вЂњaвЂќ is negative
Only cube root of вЂ“ 8: 3 пЂ­ 8 пЂЅ пЂ­ 2
3
6
Only cube root of 6:
No such thing
as a principle
cube root!
Odd Roots (3,5,7,вЂ¦)
вЂў The odd nth root of a real number вЂњaвЂќ is a
real number that multiplies by itself вЂњnвЂќ
times to give вЂњaвЂќ
вЂў Every real number вЂњaвЂќ has exactly one
odd root that is positive when вЂњaвЂќ is
positive, and negative when вЂњaвЂќ is negative
The only fifth root of - 32: 3 пЂ­ 32 пЂЅ пЂ­ 2
The only fifth root of -7:
пЂ­5 7
Rational, Irrational, and Non-real
вЂў
n
a is non-real only if the radicand is negative and the
index is even
6
вЂў
n
пЂ­ 20 is non - real because radicand
and index is even
only if the radicand
a represents a rational number
th
can be written as a вЂњperfect n вЂќ power of an integer or
the ratio of two integers
5
вЂў
is negative
n
пЂ­ 32 is rational because пЂ­ 32 пЂЅ пЂЁпЂ­ 2 пЂ©
5
5
пЂ­ 32 пЂЅ пЂ­ 2
a represents an irrational number only if it is a real
number and the radicand can not be written as вЂњperfect
nthвЂќ power of an integer or the ratio of two integers
.
4
8 is irrational
of an integer
because 8 is not the fourth power
or the ratio of two integers
4
8
Homework Problems
вЂў Section: 10.1
вЂў Page: 666
вЂў Problems: All: 1 вЂ“ 6, Odd: 7 вЂ“ 31,
39 вЂ“ 57, 65 вЂ“ 91
вЂў MyMathLab Homework Assignment 10.1
for practice
вЂў MyMathLab Quiz 10.1 for grade
Exponential Expressions
an
вЂњaвЂќ is called the base
вЂњnвЂќ is called the exponent
вЂў If вЂњnвЂќ is a natural number then вЂњanвЂќ means that вЂњaвЂќ is to be multiplied
by itself вЂњnвЂќ times.
Example: What is the value of 24 ?
(2)(2)(2)(2) = 16
вЂў 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
вЂў Meanings of exponents that are not natural numbers will be
discussed in this unit.
Negative Exponents: a-n
вЂў A negative exponent has the meaning:
вЂњreciprocate the base and make the exponent
positiveвЂќ
n
a
пЂ­n
пѓ¦1пѓ¶
пЂЅпѓ§ пѓ·
пѓЁaпѓё
Examples:
2
3
.
пЂ­2
пѓ¦2пѓ¶
пѓ§ пѓ·
пѓЁ3пѓё
1
пѓ¦1пѓ¶
пЂЅ пѓ§ пѓ· пЂЅ
9
пѓЁ3пѓё
пЂ­3
3
27
пѓ¦3пѓ¶
пЂЅ пѓ§ пѓ· пЂЅ
8
пѓЁ2пѓё
Quotient Rule for Exponential
Expressions
вЂў 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
a
m
a
n
5
4
5
7
пЂЅa
mпЂ­n
Examples:
.
x
пЂЅ 5
4пЂ­7
пЂЅ5
пЂ­3
12
x
4
пЂЅ
x
12 пЂ­ 4
пЂЅ x
8
Rational Exponents (a1/n)
and Roots
1
вЂў An exponent of the form n
has the meaning: вЂњthe nth root of the base, if it
exists, and, if there are two nth roots, it means
the principle (positive) oneвЂќ
1
a
n
, if it exists, is the n
th
root of a
1
(If there are two n
th
roots,
a n is the principle
(positive)
1
( a n multiplies
by itself
n times
to give a )
one)
Examples of
Rational Exponent of the Form:
1/n
1
100
2
пЂЅ 10 (positive
square root of 100)
1
52 пЂЅ
1
пЂЁпЂ­ 3 пЂ© 2
5 (positive
square root of 5)
пЂЅ (Does not exist! )
1
пЂ­ 32 пЂЅ пЂ­
3 (negative
square root of 3)
1
74 пЂЅ
1
пЂЁпЂ­ 9 пЂ© 7
1
пЂЁпЂ­ 8 пЂ© 6
пЂЅ
пЂЅ
4
7
7 (positive
пЂ­ 9 (seventh
fourth root of 7)
root of negative
(Does not exist! )
9)
.
Meaning of a1/n
вЂў When n is odd:
вЂ“ a1/n always exists and is either positive,
negative or zero depending on whether вЂњaвЂќ is
positive, negative or zero
вЂў When n is even:
вЂ“ a1/n never exists when вЂњaвЂќ is negative
вЂ“ a1/n always exists and is positive or zero
depending on whether вЂњaвЂќ is positive or zero
Rational Exponents of the Form:
m/n
вЂў An exponent of the form m/n has two equivalent
meanings:
(1)
am/n means find the nth root of вЂњaвЂќ, then raise
it to the power of вЂњmвЂќ
(assuming that the nth root of вЂњaвЂќ exists)
(2)
am/n means raise вЂњaвЂќ to the power of вЂњmвЂќ
then take the nth root of am
(assuming that the nth root of вЂњamвЂќ 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
вЂў 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 пѓ—37 пЂЅ 37
2
Quotient Rule:
am/an = am-n
37
4
пѓ—пЂЅ 3
пЂ­
2
7
37
Negative Exponents:
a-n = (1/a)n
4
пЂ­
3
.
4
7
пѓ¦1пѓ¶
пЂЅ пѓ§ пѓ·
пѓЁ3пѓё
7
Definitions and Rules for
Exponents
2
Power Rules:
пѓ¦ 4
пѓ§37
пѓ§
пѓЁ
(am)n = amn
(ab)m = ambm
(a/b)m
Zero Exponent:
.
=
am /
bm
a0 = 1 (a not zero)
8
пѓ¶7
пѓ· пЂЅ 3 49
пѓ·
пѓё
2
2
пЂЁ3 x пЂ© 7
2
пЂЅ 37 x 7
2
2
37
пѓ¦ 3 пѓ¶7
пѓ§ пѓ· пЂЅ
2
пѓЁ4пѓё
47
0
пѓ¦3пѓ¶
пѓ§ пѓ· пЂЅ 1
пѓЁ4пѓё
вЂњSlide RuleвЂќ for Exponential
Expressions
вЂў 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:
m
пЂ­n
пЂ­r
s
a b
c d
r
пЂЅ
c d
a
пЂ­m
пЂ­s
b
n
вЂў This rule applies to all types of exponents
вЂў Often used to make all exponents positive
Simplifying Products and
Quotients Having Factors with
Rational Exponents
вЂў All factors containing a common base can be
combined using rules of exponents in such a way
that all exponents are positive:
вЂў Use rules of exponents to get rid of parentheses
вЂў Simplify top and bottom separately by using product
rules
вЂў Use slide rule to move all factors containing a common
base to the same part of the fraction
вЂў If any exponents are negative make a final application of
the slide rule
Simplify the Expression:
пѓ¦
пѓ¶
3
8пѓ§ y y пѓ·
пѓ§
пѓ·
пѓЁ
пѓё
1
3
2
пЂ­1
6
y
пЂ­2
пЂ­
2
3
1
7
6
пЂ­
y
9
пЂ­1
y 12
2 пѓ—8
1
y4y
8y
пЂ­1
16
39
y 12
3
пЂ­
2
2
3
пЂ­1
7
1
пЂ­
8
3
8y
y y
8y
2
пЂ­
4
пЂ­
2
пЂ­2
y 12 y
8
y 12 y 3
6
16
3
пЂ­
2
12
7
32
y 12 y 12
Applying Rules of Exponents
in Multiplying and Factoring
пѓ¦ пЂ­1
пѓ¶пѓ¦ 1
пѓ§ x 2 пЂ­ 2 пѓ·пѓ§ x 2 пЂ« x
пѓ§
пѓ·пѓ§
пѓЁ
пѓёпѓЁ
вЂў Multiply:
пЂЅ x
пЂ­
1
2
пЂ«
1
2
пЂ«x
пЂ­
1
пЂ­
2
1
1
2
пЂ­ 2x 2 пЂ­ 2x
пЂ­
пЂ­
1
2
пЂ­
пЂ­
пЂ­
пЂ­
пѓ¶
2 2
2
2
2
пѓ· пЂЅ x x пЂ« x x пЂ­ 2x пЂ­ 2x 2
пѓ·
пѓё
1
1
1
1
1
2
1
пЂЅ x пЂ«x
0
пЂ­1
пЂ­ 2x 2 пЂ­ 2x
1
пЂЅ 1пЂ« x
пЂ­1
пЂ­ 2x2 пЂ­ 2x
вЂў Factor out the indicated factor:
пЂЅ x
пЂ­
3
4
пЂЁ __ пЂ« __ пЂ©
4
пѓ¦
пѓ¶
4пѓ§
4 пѓ·
пЂЅ x
5пЂ« x
пѓ§
пѓ·
пѓЁ
пѓё
пЂ­
3
пЂЅ x
пЂ­
3
4
пЂ­
5x
пЂЁ5 пЂ« x пЂ©
пЂ­
пЂ­
1
1
1
2
1
2
3
1
4
пЂ« x4 ; x
пЂ­
3
4
вЂў Roots of real numbers may be indicated
by means of either rational exponent
notation or radical notation:
n
a is called a RADICAL
is called a RADICAL
(expressio n)
SIGN
n is called the INDEX
a is called the RADICAND
вЂў
вЂў
вЂў
вЂў
вЂў
If no index is shown it is assumed to be 2
When index is 2, the radical is called a вЂњsquare rootвЂќ
When index is 3, the radical is called a вЂњcube rootвЂќ
When index is n, the radical is called an вЂњnth rootвЂќ
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
вЂў 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
вЂў
An exponential expression with exponent of the form
вЂњm/nвЂќ can be converted to radical notation with index of
вЂњnвЂќ, and vice versa, by either of the following formulas:
m
1.
a
n
2
пЂЅ
m
пЂЅ
n
a
8 пЂЅ
m
n
m
пЂЅ
2
3
3
2
8 пЂЅ
8
3
2
3
64 пЂЅ 4
пЂЅ пЂЁ2 пЂ© пЂЅ 4
2
2.
a
вЂў
These definitions assume that the nth root of вЂњaвЂќ exists
n
3
Examples
4
5 пЂЅ
7
4
7
OR
7
5
4
9
5
8 пЂЅ 85
9
3
4x
11
пЂЅ 4
11
x
3
OR 4
11
3
.
n
x
n
.
вЂў If вЂњnвЂќ is even, then this notation means
principle (positive) root:
n
x
пЂЅ x
n
(absolute
value needed to insure positive
вЂў If вЂњnвЂќ is odd, then:
n
x
n
x
n
пЂЅ x
вЂў If we assume that вЂњxвЂќ is positive (which
we often do) then we can say that:
n
пЂЅ x
.
Homework Problems
вЂў Section: 10.2
вЂў Page: 675
вЂў Problems: All: 1 вЂ“ 10, Odd: 11 вЂ“ 47,
51 вЂ“ 97
вЂў MyMathLab Homework Assignment 10.2
for practice
вЂў MyMathLab Quiz 10.2 for grade
Product Rule for Radicals
вЂў 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
n
a b пЂЅ
n
n
ab
4
3 5 пЂЅ 3 пѓ— 5 пЂЅ 15
4
4
4
вЂў This rule works both directions:
n
ab пЂЅ
n
a
n
b
3
16 пЂЅ
3
8 2 пЂЅ 2 2
3
3
Quotient Rule for Radicals
вЂў 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
4
n
5
5
a
a
пЂЅ 4
пЂЅn
4
n
7
b
7
b
вЂў This rule works both directions:
n
.
a
b
n
пЂЅ
a
3
n
b
5
8
3
пЂЅ
3
5
8
3
пЂЅ
5
2
Root of a Root Rule for Radicals
вЂў When you take the mth root of the nth root of
a radicand вЂњaвЂќ, it is the same as taking a
single root of вЂњaвЂќ using an index of вЂњmnвЂќ
.
m n
a пЂЅ
4 3
6 пЂЅ
mn
12
a
6
NO Similar Rules for Sum and
.
n
aпЂ«
b п‚№
n
27 пЂ«
3
n
n
a пЂ­n b п‚№
3
aпЂ«b
8 п‚№
3
35
3пЂ« 2 п‚№
3
35
n
3
aпЂ­b
27 пЂ­ 3 8 п‚№
3
19
3пЂ­2 п‚№
3
19
вЂў
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 вЂњrationalizedвЂќ)
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 3
3
2
2
2. Simplify the first radical
3
2 2
Example
Problem?
3
3
3
2
5
3
2
5
3
2 y
3 3
2y
3
24 x y
2 3x y
Is there another exponent t
hat is too big?
Write this as a product of two radicals :
2
3x y
2
3x y
2
2
Simplify t
he first radical :
Simplifying when a Denominator
Contains a Single Radical of
Index вЂњnвЂќ
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 вЂњ1вЂќ
where 1 is in the form of: n m
n
m
and m is the product of all the factors required
make every exponent
3.
to
be equal to "n"
Simplify to eliminate the radical in the denominator
Example
3
5
3
пЂЅ
4x y
пЂЅ
3
6
5
5
пѓ—
3
пЂЅ
3
2 x y
3
2
2
y5 2 x y
5
3
6
3
5
2
2 x y
3
2
2 x y
y
5 5
4
пЂЅ
4
3
пЂЅ
2
3
2
3
y5 2 x y
2 x y
3
2
4
5
5
5
35 2 x y
y5 2 x y
пЂЅ
5
пЂЅ
3
2
2 xy
2
35 2 x y
2
4
3 8x y
2 xy
2
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
пЂЅ
5
3
4
5
пЂЅ
5
3
2
2
5
пЂЅ
5
3
2
2
5
пѓ—
5
2
3
2
3
5
пЂЅ
3пѓ—2
5
5
пЂЅ
2
3
5
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
2 3 x y
All exponents
Dividing
3
2
8
3
all exponents
4
2 3 x y
Problem?
2
пЂЅ
3
and index share what factor?
in and index by 2 gives :
3
3 x
3 3
2
2 xy
пЂЅ 3 x 3 4 xy
2
Simplifying Expressions Involving
Products and/or Quotients of
Radicals with the Same Index
вЂў Use the product and quotient rules to
combine everything under a single
вЂў Simplify the single radical by procedures
previously discussed
Example
4
3 4
ab
4
ab
3
a b
4
пЂЅ
4
3
3
a b
a
4
2
пЂЅ
4
a b
4
3
3
a b
4
пЂЅ
пЂЅ
a
3
a b
a
4
b
4
пЂЅ
4
b
a
пЂЅ
4
b
4
a
3
4
a
4
a
3
Right Triangle
вЂў A вЂњright triangleвЂќ is a triangle that has a 900
angle (where two sides intersect
perpendicularly)
b
90
0
a
вЂў The side opposite the right angle is called
the вЂњhypotenuseвЂќ and is traditionally
identified as side вЂњcвЂќ
вЂў The other two sides are called вЂњlegsвЂќ and
are traditionally labeled вЂњaвЂќ and вЂњbвЂќ
Pythagorean Theorem
вЂў In a right triangle, the square of the
hypotenuse is always equal to the sum of
the squares of the legs:
c пЂЅ a пЂ«b
2
c
b
90
0
a
2
2
Pythagorean Theorem Example
вЂў 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
90
0
4
вЂў Note that Pythagorean Theorem is true:
c пЂЅ a пЂ«b
2
2
2
5 пЂЅ 4 пЂ«3
25 пЂЅ 16 пЂ« 9
2
2
2
Using the Pythagorean Theorem
вЂў 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
вЂў Given two sides of a right triangle with one
side unknown:
вЂ“ Plug two known values and one unknown
value into Pythagorean Theorem
вЂ“ Use addition or subtraction to isolate the
вЂњvariable squaredвЂќ
вЂ“ Square root both sides to find the desired
Example
вЂў Given a right triangle with a пЂЅ 7 and c пЂЅ 25
find the other side.
c пЂЅ a пЂ«b
2
2
2
25 пЂЅ 7 пЂ« b
2
625 пЂЅ 49 пЂ« b
2
2
2
625 пЂ­ 49 пЂЅ 49 пЂ­ 49 пЂ« b
576 пЂЅ b
24 пЂЅ
576 пЂЅ b
2
2
Homework Problems
вЂў Section: 10.3
вЂў Page: 685
вЂў Problems: Odd: 7 вЂ“ 19, 23 вЂ“ 57,
61 вЂ“ 107
вЂў MyMathLab Homework Assignment 10.3
for practice
вЂў MyMathLab Quiz 10.3 for grade
вЂў Addition and subtraction of radicals can always be
indicated, but can be simplified into a single radical
only when the radicals are вЂњlike radicalsвЂќ
вЂў вЂњLike RadicalsвЂќ 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
вЂў When вЂњlike radicalsвЂќ are added or subtracted, the
result is a вЂњlike radicalвЂќ with coefficient equal to the
sum or difference of the coefficients
3
4
5 пЂ«2
-2
4
5 пЂ« 3
5 пЂЅ 5
4
3
5 пЂЅ
4
5
Okay as is - can' t combine
unlike
Note Concerning Adding and
вЂў When addition or subtraction of
radicals is indicated you must first
simplify all radicals because some
radicals that do not appear to be like
simplified
Example
Not like terms (yet)
3
128 пЂ­ 5
пЂЅ
3
3
2 2
3 3
2пЂ«2
3
2 пЂ­5
3
пЂЅ 43 2 пЂ­ 5
3
2пЂ«4
пЂЅ
16
2пЂ«2
3
Simplify
3
3
3
2 пЂ­5
7
3 3
2
пЂЅ3
3
2
individual
2пЂ«2
3
2
4
пЂЅ 2 пѓ— 23 2 пЂ­ 5
:
2
3
2
3
2 пЂ« 2пѓ—2
3
2
Homework Problems
вЂў Section: 10.4
вЂў Page: 691
вЂў Problems: Odd: 5 вЂ“ 57
вЂў MyMathLab Homework Assignment 10.4
for practice
вЂў 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 вЂњ1вЂќ where вЂњ1вЂќ is in the
form of a вЂњspecial radicalвЂќ over itself
3. The вЂњspecial radicalвЂќ 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
2
3
Simplify
denominato
r:
9x
3
2
1
2
3
3 x
3
3
2
2
3
3 x
3
2 пѓ—3x
3
3
3
3 x
3
3x
3x
Multiply
by special "1" :
6x
3x
2
2
3
Use product rule :
2
Simplify
denominato
r:
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 вЂњ1вЂќ where вЂњ1вЂќ is in the
form of a вЂњspecial binomial radicalвЂќ over
itself
3. The вЂњspecial binomial radicalвЂќ is the conjugate
of the denominator (same terms вЂ“ opposite
sign)
4. Complete multiplication (the denominator will
Example
5
3пЂ«
3пЂ«
Multiply
fraction
3пЂ­
2
Distribute
3пЂ­
2
FOIL on bottom
2
5
пѓ—
2
r doesn' t need simplifyin
by special one :
on top :
:
15 пЂ­ 10
9пЂ­
4
Simplify
bottom :
15 пЂ­ 10
3пЂ­2
15 пЂ­ 10
g
Homework Problems
вЂў Section: 10.5
вЂў Page: 700
вЂў Problems: Odd: 7 вЂ“ 105
вЂў MyMathLab Homework Assignment 10.5
for practice
вЂў MyMathLab Quiz 10.5 for grade
вЂў An equation is called a radical equation if it
contains a variable in a radicand
вЂў Examples:
3
xпЂ­
xпЂ­3 пЂЅ5
xпЂ«
xпЂ«5 пЂЅ1
x пЂ« 4 пЂ­ 3 2x пЂЅ 0
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 вЂњapparent
solutionsвЂќ
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?
вЂў Raising both sides of an equation to a power does not always result
in equivalent equations
вЂў If both sides of equation are raised to an odd power, then resulting
equations are equivalent
вЂў If both sides of equation are raised to an even power, then resulting
equations are not equivalent (вЂњextraneous solutionsвЂќ may be
introduced)
вЂў Raising both sides to an even power, may make a false statement
true:
2
2
4
4
пЂ­ 2 п‚№ 2 , however
: пЂЁ- 2 пЂ© пЂЅ пЂЁ 2 пЂ© , пЂЁ- 2 пЂ© пЂЅ пЂЁ 2 пЂ© , etc.
вЂў Raising both sides to an odd power never makes a false statement
true:
3
3
5
5
пЂ­ 2 п‚№ 2 , and : пЂЁ- 2 пЂ© п‚№ пЂЁ 2 пЂ© , пЂЁ- 2 пЂ© п‚№ пЂЁ 2 пЂ© , etc.
.
Example of Solving
xпЂ­
xпЂ­3 пЂЅ5
xпЂ­5пЂЅ
Check x пЂЅ 4
2
пЂЅ
4пЂ­
xпЂ­3
пЂЁ
xпЂ­3
2
x пЂ­ 10 x пЂ« 25 пЂЅ x пЂ­ 3
2
x пЂ­ 11 x пЂ« 28 пЂЅ 0
2
пЂЁ x пЂ­ 4 пЂ©пЂЁ x пЂ­ 7 пЂ© пЂЅ 0
x пЂ­ 4 пЂЅ 0 OR
xпЂ­7пЂЅ0
x пЂЅ 4 OR x пЂЅ 7
4пЂ­3 пЂЅ5?
4пЂ­ 1пЂЅ5?
3п‚№ 5
x пЂЅ 4 is NOT a solution
Check x пЂЅ 7
7пЂ­
7пЂ­
7пЂ­3 пЂЅ5?
4 пЂЅ5?
5пЂЅ5
xпЂЅ7
IS a solution
Example of Solving
xпЂ«
пЂЁ
x пЂ« 5 пЂЅ 1пЂ­
xпЂ«5
Check x пЂЅ 4
xпЂ«5 пЂЅ1
2
пЂЁ
x
пЂЅ 1пЂ­
x
4пЂ«
2
x пЂ« 5 пЂЅ 1пЂ­ 2 x пЂ« x
2
пЂЅ
9 пЂЅ1?
2пЂ«3 пЂЅ1?
x пЂЅ 4 is NOT a solution
x
4пЂЅ x
4пЂ«
5п‚№1
4 пЂЅ пЂ­2 x
пЂ­2пЂЅ
4пЂ«5 пЂЅ1?
2
Equation
has No Solution!
пЃ¦
Example of Solving
3
x пЂ« 4 пЂ­ 3 2x пЂЅ 0
3
пЂЁ
3
xпЂ«4 пЂЅ
xпЂ«4
3
3
3
2x
2x
x пЂ« 4 пЂЅ 2x
4пЂЅ x
(No need to check)
3
Homework Problems
вЂў Section: 10.6
вЂў Page: 709
вЂў Problems: Odd: 7 вЂ“ 57
вЂў MyMathLab Homework Assignment 10.6
for practice
вЂў MyMathLab Quiz 10.6 for grade
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