Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations.
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Transcript of Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations.
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?What is another square root of 9?
• 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 signsThe two square roots of 16 are:The two square roots of 5 are:
33
ROOT) PRINCIPLE :Root Square (Positive
16 and 16 5 and 5
4 and 4 :simplified
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
• Every positive real number “a” has two even roots that have equal absolute values, but opposite signsThe fourth roots of 16: The fourth roots of 7:
ROOT) PRINCIPLE :RootEven (Positive
44 7- and 72 and 2 :simplified
existnot does 164
44 16 and 16
Radical Expressions
• On the previous slides we have used symbols of the form:
• This is called a radical expression and the parts of the expression are named:
Index:
Radical Sign :
Radicand:
• Example:
n a
n
a
5 8 8:Radicand 5:Index
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:
Only cube root of 6:
23 6
root! cube principle a as such thing No
3 8
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 negativeThe only fifth root of - 32:
The only fifth root of -7:
3 32
5 7
2
Rational, Irrational, and Non-real Radical Expressions
• is non-real only if the radicand is negative and the index is even
• 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
• 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
.
n a
5
232 because rational is 325
n a
n a2325
even isindex and negative is radicand because real-non is 206
integers twoof ratio or theinteger an of
powerfourth not the is 8 because irrational is 84
4 8
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 Expressionsan
“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) = - 81Example: 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”
Examples:
.
nn
aa
1
23
3
3
2
9
1
3
12
8
27
2
33
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
Examples:
.
nmn
m
aa
a
7
4
5
5
4
12
x
x
374 55
8412 xx
Rational Exponents (a1/n)and Roots
• 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”
n
1
aa n ofroot n theis exists,it if , th1
one) (positive) principle theis roots, n twoare there(If1
th na
) give ton times itselfby multiplies (1
aa n
Examples ofRational Exponent of the Form:
1/n
.
2
1
100 10 100) ofroot square (positive
2
1
5 5 5) ofroot square (positive
2
1
3 )exist!not (Does
2
1
3 3 3) ofroot square (negative
4
1
7 7) ofroot fourth (positive4 7
7
1
9 7 9 9) negative ofroot (seventh
6
1
8 )exist!not (Does
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:Product Rule: aman = am+n
Quotient Rule: am/an = am-n
Negative Exponents: a-n = (1/a)n
.
7
2
7
4
33 7
6
3
7
4
7
2
3
37
2
3
7
4
37
4
3
1
Definitions and Rules for Exponents
Power Rules:(am)n = amn
(ab)m = ambm
(a/b)m = am / bm
Zero Exponent: a0 = 1 (a not zero)
.
7
2
7
4
3 49
8
3
7
2
3x 7
2
7
2
3 x
7
2
4
3
7
2
7
2
4
3
0
4
31
“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 exponentExample: Use rule to slide all factors to other part of the fraction:
• This rule applies to all types of exponents• Often used to make all exponents positive
sr
nm
dc
banm
sr
ba
dc
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:
6
1
4
31
2
3
1
2
8
yy
yy
6
1
4
31
23
2
2
8
yy
yy
12
2
12
91
3
6
3
2
2
8
yy
yy
12
71
3
8
2
8
y
y
3
8
12
7
1 82
yy
12
32
12
7
16
yy
12
39
16
y
Applying Rules of Exponentsin Multiplying and Factoring
• Multiply:
• Factor out the indicated factor:
2
1
2
1
2
1
2 xxx 2
1
2
1
2
1
2
1
2
1
2
1
22
xxxxxx
2
1
2
1
2
1
2
1
2
1
2
1
22
xxxx 2
1
2
110 22
xxxx
2
1
2
11 221
xxx
4
3
4
1
4
3
;5
xxx
____4
3
x xx
54
3
4
4
4
3
5 xx
Radical Notation
• Roots of real numbers may be indicated by means of either rational exponent notation or radical notation:
n)(expressio RADICAL a called is n a
INDEX thecalled is n
SIGN RADICAL a called is
RADICAND thecalled is a
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
• 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:
1.
2.
• These definitions assume that the nth root of “a” exists
n mn
m
aa
3
2
8
3
2
8
mnn
m
aa 42 2
4643 3 28
23 8
Examples
.
7
4
5 47 5
5 98 5
9
8
11
3
4x 11 34 x
7 45 OR
3114 OR x
.
• If “n” is even, then this notation means principle (positive) root:
• If “n” is odd, then:
• If we assume that “x” is positive (which we often do) then we can say that:
.
n nx
xxn n
xxn n
xxn n
answer) positive insure toneeded value(absolute
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:
• This rule works both directions:
nnn abba
nnn baab
44 53 4 53 4 15
3 16 33 28 3 22
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
• This rule works both directions:
.
nn
n
b
a
b
a
n
n
n
b
a
b
a
3
3
8
5
4
4
7
54
7
5
3
8
5
2
53
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”
.
mnm n aa
4 3 6 12 6
NO Similar Rules for Sum and Difference of Radicals
.
nnn baba
nnn baba
35827 333 3523 3
19827 333
1923 3
Simplifying Radicals
• A radical must be simplified if any of the following conditions exist:
1. Some factor of the radicand has an exponent that is bigger than or equal to the index
2. There is a radical in a denominator (denominator needs to be “rationalized”)
3. The radicand is a fraction4. All of the factors of the radicand have
exponents that share a common factor with the index
Simplifying when Radicand has Exponent Too Big
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
2. Simplify the first radical
3 42
33 3 22
3 22
Example
3 5224 yx
3 52332 yx
3 223 33 32 yxy
3 2232 yxy
big? tooishat exponent tanother thereIs
Problem?
:radicals twoofproduct a as thisWrite
:radicalfirst heSimplify t
Simplifying when a Denominator Contains a Single Radical of
Index “n”1. Simplify the top and bottom separately to get rid of
exponents under the radical that are too big2. Multiply the whole fraction by a special kind of “1”
where 1 is in the form of:
3. Simplify to eliminate the radical in the denominator
n
n
m
m
"n"
m
toequal be radicand in theexponent every make
torequired factors theall ofproduct theis and
Example
5 634
3
yx 5 6322
3
yx
2
5 42
2
83
xy
yx
5 325 5 2
3
yxy
5 322
3
yxy
5 423
5 423
5 32 2
2
2
3
yx
yx
yxy
5 555
5 423
2
23
yxy
yx
2
5 423
2
23
xy
yx
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
4
35
5
4
3
5 2
5
2
3
5 3
5 3
5 2
5
2
2
2
3
5 5
5 3
2
23
2
245
Simplifying when All Exponents in Radicand Share a Common
Factor with Index1. Divide out the common factor from the index
and all exponents
Problem?
6 286432 yx
3 43232 yx
factor? what shareindex and radicandin exponents All 2:gives 2by index andin exponents all Dividing
3 23 33 23 xyx 3 43 xyx
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 33
44 3
ba
abab4
33
42
ba
ba 4
a
b
4
4
a
b
4 3
4 3
4
4
a
a
a
b
4 4
4 3
a
ba
a
ba4 3
Right Triangle
• A “right triangle” is a triangle that has a 900 angle (where two sides intersect perpendicularly)
• 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”
090
hypotenusecba
Pythagorean Theorem
• In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs:
090
cb
a
222 bac
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.
• Note that Pythagorean Theorem is true:
090
53
4
222 bac 222 345
91625
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 find the other side.
25 and 7 ca
222 bac 222 725 b249625 b
2494949625 b2576 b
b57624
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 coefficientsWhich are like radicals?
• When “like radicals” are added or subtracted, the result is a “like radical” with coefficient equal to the sum or difference of the coefficients
344 53 and 52- ,54 ,53
44 5253
34 5352 -
4 55
radicals unlike combinet can' - is asOkay
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
333 16225128 3 433 7 22252
33 3333 33 22225222 333 22225222
333 242524 3 23
(yet) termslikeNot :radicals individualSimplify
:radicals like All
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
9
2
x
3 2
3
3
2
x
3 2
3 2
3 2
3
3
3
3
2
x
x
x
3 33
3 2
3
32
x
x
x
x
3
63 21:rdenominatoSimplify
:"1" specialby Multiply
:ruleproduct Use
:rdenominatoSimplify
Simplifying to Get Rid of a Binomial Denominator that Contains One or
Two Square Root Radicals1. Simplify the radical(s) in the denominator2. 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
23
5
gsimplifyin needt doesn'r denominatoin Radical
:one specialby fraction Multiply
23
23
23
5
:on top Distribute
:bottomon FOIL
49
1015
:bottomSimplify
23
1015
1015
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:
53 xx
024 33 xx
15 xx
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:
• Raising both sides to an odd power never makes a false statement true:
.
etc. ,22- ,22- :however , 22 4422
etc. ,22- ,22- :and , 22 5533
Example of SolvingRadical Equation
53 xx
35 xx
22 35 xx
325102 xxx
028112 xx 074 xx
07 OR 04 xx7 OR 4 xx
4xCheck
?5344 ?514
53
7xCheck
?5377 ?547
55
solution a NOT is 4x
solution a IS 7x
Example of SolvingRadical Equation
15 xxxx 15
2215 xx
xxx 215
x24 x 2
222 x
x4
4xCheck
?1544 ?194
?132 15
solution a NOT is 4x
Solution! No hasEquation
Example of SolvingRadical Equation
024 33 xx33 24 xx
33
33 24 xx
xx 24 x4
check) toneed (No
Homework Problems
• Section: 10.6
• Page: 709
• Problems: Odd: 7 – 57
• MyMathLab Homework Assignment 10.6 for practice
• MyMathLab Quiz 10.6 for grade