4.4 Fractional Exponents and Radicals. Construct Understanding.
Radicals Exps, Rational Exponents, Set of Complex Nos

Upload
kimberlyrosemallari 
Category
Documents

view
222 
download
0
Transcript of Radicals Exps, Rational Exponents, Set of Complex Nos

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
1/30
Radicals

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
2/30
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

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
3/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
4/30
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 ).

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
5/30
• 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

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
6/30
• 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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
7/30
• 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%

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
8/30
• 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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
9/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
10/30
*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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
11/30
• Examples – 1) 4)
–
– 2) 5)
–
– 3)

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
12/30
• 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 aproblemb#problem basis" we shall merel# assume that all$ariables represent positi$e real numbers.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
13/30
Relationship betweenexponents and roots

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
14/30
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

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
15/30
• 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

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
16/30
• 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

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
17/30
• Example
• 1) *)
• 2) +)
• 3)
• 4) ,)
• 5)
• 6)

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
18/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
19/30
Example1) 5)
2) 6)
3)
4)

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
20/30
omplex .mbers

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
21/30
/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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
22/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
23/30
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

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
24/30
Example1)
2)
3)

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
25/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
26/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
27/30
Example1) 6)
2)
3)
4)
5)

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
28/30
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.

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
29/30
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%

8/19/2019 Radicals Exps, Rational Exponents, Set of Complex Nos
30/30
Examples:1) 3i % 52i)
2) 2 – 3i)%4*i)
3) 45i)%2i