Radicals Exps, Rational Exponents, Set of Complex Nos

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos

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Radicals


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Recall from our work with exponents that to square a numbermeans to raise it to the second power – that is to use the numberas a factor twice.

A square root of a number is one of its two equal factors. Thus 4and -4 are both square roots of 16

n !eneral" a is a square root of b is


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The s#mbol " called a radical si!n" is used to desi!nate thenonne!ati$e square root" which is called the principal squareroot. The number under the radical si!n is called the radicand"and the entire expression" such as " is referred to as a radical.


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n !eneral" if n is an e$en positi$e inte!er" then thefollowin! statements are true%

1&'$er# positi$e real number has exactl# two real nthroots" one positi$e and one ne!ati$e. (or example"the real fourth roots of 16 are ) and -))&*e!ati$e real numbers do not ha$e real nth roots.(or example" there are no real fourth roots of -16.

n !eneral" if n is an odd positi$e inte!er !reater than1" then the followin! statements are true.

1&'$er# real number has exactl# one real nth root.)&The real nth root of a positi$e number is positi$e.(or example" the +fth root of ,) is ).,&The real nth root of a ne!ati$e number is ne!ati$e.

(or example" the +fth root of -,) is ).


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• Examples

e+nition 1%

if and onl# if

f n is an e$en positi$e inte!er" then a and b are both

nonne!ati$e. f n is an odd positi$e inte!er !reater than 1" then aand b are both nonne!ati$e or both ne!ati$e. The s#mboldesi!nates the principal root.

To complete our terminolo!#" the n in the radical is called theindex of the radical. f n ) we commonl# write


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• Examples

e+nition )%

if and are real numbers

e+nition ) statest that the nth root of a product is equal to theproduct of the nth roots.

e+nition 1 and ) pro$ides the basis for chan!in! radicals tosimplest radical form. This means that the radicand does notcontain an# perfect powers of the index.


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• Examples – 1)

– 2)

– 3)

– 4)

– 5)

– 6)

The distributi$e propert# can be used to combine radicals thatha$e the same index and the same radicand%


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• Example• 1) 2) 3) 4)

e+nition ,%

if and are real numbersand c is not equal to /.

e+nition , states that the nth root of a quotient is equal to thequotient of the nth roots.


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0efore we consider more examples" lets summari2esome ideas about simplif#in! radicals. A radical issaid to be in simplest radical form if the followin!conditions are satis+ed%

1&*o fractions appears within a radical si!n.)&*o radical appears in the denominator.,&*o radicand contains a perfect power of the index.


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*ow lets consider an example in which neither the numerator northe denominator of the radicand is a perfect nth power%

The process used to simplif# the radical in this example isreferred to as rationali2in! the denominator. There is more thanone wa# to rationali2e the denominator" as illustrated b# the nextexample.


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• Examples – 1) 4)

–

– 2) 5)

–

– 3)


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• Examples:

– 1) 4)

– 2)

– 3)

'$er# radical expression with $ariables in the radicand needs tobe anal#2ed indi$iduall# to determine the necessar# restrictionson the $ariables. 3owe$er" to a$oid ha$in! to do this on aproblem-b#-problem basis" we shall merel# assume that all$ariables represent positi$e real numbers.


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Relationship betweenexponents and roots


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From or std! o" radicals we #now that:

$" is to hold when m is a rationalnmber o" the "orm 1%p& where p is apositi'e inte(er (reater than 1 and n p&then

et is consider the followin! comparisons


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• Example

• 1)

• 2)

• 3)

• 4)

e+nition 4%

f b is a real number" n is a positi$e inte!er !reaterthan 1" and exists" then


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• Example

• 1)

• 2)

e+nition 5%

f mn is a rational number expressed in lowest terms"where n is a positi$e inte!er !reater than 1" and m isan# inte!er" and if b is a real number such that

exists" then


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• Example

• 1) *)

• 2) +)

• 3)

• 4) ,)

• 5)

• 6)


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Example

The link between exponents and roots pro$ides a basis formultipl#in! and di$idin! some radicals e$en if the# ha$edi7erent indexes.

The !eneral procedure is to chan!e from radical to exponentialform" appl# the properties of exponents" and then chan!e back

to radical form.


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Example1) 5)

2) 6)

3)

4)


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-omplex .mbers


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/et0s be(in b! de"inin( a nmber i sch that

There are some $er# simple equations that do not ha$e solutionsif we restrict oursel$es to the set of real numbers. (or example"equation x8) 91 / has no solutions amon! the real numbers.

The number i is not a real number and is often called the

ima!inar# unit" but the number i8) is the real number -1.


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he a bi is called the standard form o" a complex nmber

he real nmber a is called the real part o" the complexnmber& and&

b is called the imaginary part .ote that b is a real nmber)

Example:

he set o" real nmbers is a sbset o" complex nmbers

e+nition 6

A complex number is an# number that can beexpressed in the form a 9 bi

where a and b are real numbers" and i is theima!inar# unit.


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Examples:

1)43i) 5,i) 4)2)6 4i) +*i)

3)

Two complex number a 9 bi and c 9 di are said to be equal if andonl# if a c and b d. n other words" two complex numbers areequal if and onl# if their real parts are equal and their ima!inar#parts are equal.e+nition : – Addition and ;ubtraction of


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Example1)

2)

3)


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Examples:

1)

2)

3)

e+nition > – The principal square root of –b andde+ne it to be%

where b is an# positi$e real number.


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-orrect

$ncorrect

?e must be careful with the use of the s#mbol " where b@/.;ome properties that are true in the set of real numbersin$ol$in! the square root s#mbol do not hold if the square roots#mbol does not represent a real number. (or example"

does not hold if a and b are both ne!ati$e numbers.

To a$oid dicult# with this idea" #ou should rewrite allexpressions of the form " where b@/" in the formbefore doin! an# computations.


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Example1) 6)

2)

3)

4)

5)


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Examples:1) 2 3i)45i)

2)

3)

0ecause complex numbers ha$e a binomial form" we can +nd theproduct of two complex numbers in the same wa# that we +ndthe product of two binomials. Then" b# replacin! i8) with -1 wecan simplif# and express the +nal product in the standard form ofa complex number.


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The complex number )9,i and )-,i are called conBu!ates of eachother. n !eneral" the two complex numbers a 9 bi and a – bi arecalled conBu!ates of each other" and the product of a complexnumber and its conBu!ate is a real number. This can be shown asfollows%


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Examples:1) 3i % 52i)

2) 2 – 3i)%4*i)

3) 45i)%2i

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Radicals Exps, Rational Exponents, Set of Complex Nos