Rational Exponents, Radicals, and Complex Numbers

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Transcript of Rational Exponents, Radicals, and Complex Numbers
MartinGay, Intermediate Algebra, 5ed 33
Square Roots
Opposite of squaring a number is taking the square root of a number.
A number b is a square root of a number a if b2 = a.
In order to find a square root of a, you need a # that, when squared, equals a.
MartinGay, Intermediate Algebra, 5ed 44
Principal and Negative Square Roots
If a is a nonnegative number, then
is the principal or nonnegative square root of a
a
is the negative square root of a.a
Principal Square Roots
MartinGay, Intermediate Algebra, 5ed 55
Radical expression is an expression containing a radical sign.
Radicand is the expression under a radical sign.
Note that if the radicand of a square root is a negative number, the radical is NOT a real number.
Radicands
MartinGay, Intermediate Algebra, 5ed 77
Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers).
Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers.
IF REQUESTED, you can find a decimal approximation for these irrational numbers.
Otherwise, leave them in radical form.
Perfect Squares
MartinGay, Intermediate Algebra, 5ed 88
Radicands might also contain variables and powers of variables.
To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only.
1064x 58x
Perfect Square Roots
Example:
MartinGay, Intermediate Algebra, 5ed 99
Cube Root
The cube root of a real number a is written as
3 , anda
Cube Roots
abba 33 ifonly
MartinGay, Intermediate Algebra, 5ed 1111
Other roots can be found, as well.
The nth root of a is defined as
abba nn ifonly
If the index, n, is even, the root is NOT a real number when a is negative.
If the index is odd, the root will be a real number.
nth Roots
MartinGay, Intermediate Algebra, 5ed 1212
Simplify the following.
20225 ba 105ab
39
364
b
a3
4
b
a
nth Roots
Example:
MartinGay, Intermediate Algebra, 5ed 1313
nth Roots
Simplify the following. Assume that all variables represent positive numbers.
4 816x 22x
515
532
b
a
3
2
b
a3
2
b
a
Example:
MartinGay, Intermediate Algebra, 5ed 1414
If the index of the root is even, then the notation represents a positive number.
n a
But we may not know whether the variable a is a positive or negative value.
Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.
nth Roots
MartinGay, Intermediate Algebra, 5ed 1515
If n is an even positive integer, then
aan n
If n is an odd positive integer, then
aan n
Finding nth Roots
Finding n na
MartinGay, Intermediate Algebra, 5ed 1616
Simplify the following.
20225 ba 105ab 105 ba
If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign.
10202 525 abba
Finding nth Roots
MartinGay, Intermediate Algebra, 5ed 1717
39
364
b
a3
4
b
a
Simplify the following.
Since the index is odd, we don’t have to force the negative root to be a negative number.
If a or b is negative (and thus changes the sign of the answer), that’s okay.
Finding nth Roots
Example:
MartinGay, Intermediate Algebra, 5ed 1818
We can also use function notation to represent rational functions.
For example,
Evaluating a rational function for a particular value involves replacing the value for the variable(s) involved.
Evaluating Rational Functions
Example:
2( ) 4.xT xx
Find the value 2(2) 4.xT
x
222
) 42(T 4 42
2 4 2
MartinGay, Intermediate Algebra, 5ed 1919
Since every value of x that is substituted into the equation
n xy
produces a unique value of y, the root relation actually represents a function.
The domain of the root function when the index is even, is all nonnegative numbers.
The domain of the root function when the index is odd, is the set of all real numbers.
Root Functions
MartinGay, Intermediate Algebra, 5ed 2020
We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape.
You should have a basic familiarity with root functions, as well.
Root Functions
MartinGay, Intermediate Algebra, 5ed 2121
x y
4 2
1 1
0 0
x
y
(0, 0)
(4, 2)
(1, 1)
Graph xy
6 6
2 2
(2, )2(6, )6
Graphs of Root Functions
Example:
MartinGay, Intermediate Algebra, 5ed 2222
x y
1 1
0 0x
y
(0, 0)
(1, 1)
28
4 3 4 (4, )3 4(8, 2)
1 1
4 3 4
28
(1, 1)
(4, )3 4(8, 2)
Graph 3 xy
Graphs of Root Functions
Example:
MartinGay, Intermediate Algebra, 5ed 2424
Exponents with Rational Numbers
So far, we have only worked with integer exponents.
In this section, we extend exponents to rational numbers as a shorthand notation when using radicals.
The same rules for working with exponents will still apply.
MartinGay, Intermediate Algebra, 5ed 2525
Recall that a cube root is defined so that
abba 33 ifonly
However, if we let b = a1/3, then aaaab 133/133/13 )(
Since both values of b give us the same a,33/1 aa
Understanding a1/n
nn aa /1
n aIf n is a positive integer greater than 1 and is a real number, then
1Definition of na
MartinGay, Intermediate Algebra, 5ed 2626
Use radical notation to write the following. Simplify if possible.
3381 4 44 4/181
5/11032x 25 1055 10 2232 xxx
3/1716x 3233 633 743 7 2222216 xxxxxx
Using Radical Notation
Example:
MartinGay, Intermediate Algebra, 5ed 2727
Understanding am/n
If m and n are positive integers greater than 1 with m/n in lowest terms, then
mnn mnm aaa /
1
Definition of na
n aas long as is a real number
MartinGay, Intermediate Algebra, 5ed 2828
Use radical notation to write the following. Simplify if possible.
3/48 4
3 8
3/773x 3 773x
32 73)73( xx
4
3 32 42 16
33 6 7373 xx
Using Radical Notation
Example:
MartinGay, Intermediate Algebra, 5ed 2929
Understanding am/n
as long as am/n is a nonzero real number.
nmnm
aa
// 1
Definition ofm
na
MartinGay, Intermediate Algebra, 5ed 3030
Use radical notation to write the following. Simplify if possible.
3/264 3/264
1
4/516 4/516
1
23 64
1
23 34
1
24
116
1
54 42
1
52
1
32
1
Using Radical Notation
Example:
MartinGay, Intermediate Algebra, 5ed 3131
Use properties of exponents to simplify the following. Write results with only positive exponents.
33/25/132 x 25/332 x
3/2
2/14/1
a
aa 3/22/14/1a
23
5 52 x 232 x 28x
12/812/612/3a 12/11a 12/11
1
a
Using Rules for Exponents
Example:
MartinGay, Intermediate Algebra, 5ed 3232
Use rational exponents to write as a single radical.
253 2/13/1 25 6/36/2 25 6/132 25 6 200
Using Rational Exponents
Example:
MartinGay, Intermediate Algebra, 5ed 3434
n n na b ab
Product Rule for Radicals
Product Rule for Radicals
n a n bIf and are real numbers, then
MartinGay, Intermediate Algebra, 5ed 3535
Simplify the following radical expressions.
40 104 102
16
5 16
5
4
5
15 No perfect square factor, so the radical is already simplified.
Simplifying Radicals
Example:
MartinGay, Intermediate Algebra, 5ed 3636
Simplify the following radical expressions.
7x xx6 xx6 xx3
16
20
x
16
20
x
8
54
x 8
52
x
Simplifying Radicals
Example:
MartinGay, Intermediate Algebra, 5ed 3737
nn
n
a ab b
n a n bIf and are real numbers,
Quotient Rule Radicals
and is not zero, thenn b
Quotient Rule for Radicals
MartinGay, Intermediate Algebra, 5ed 3838
Simplify the following radical expressions.
3 16 3 28 33 28 3 2 2
3
64
3 3
3
64
3
4
33
Simplifying Radicals
Example:
MartinGay, Intermediate Algebra, 5ed 3939
The distance d between two points (x1,y1) and (x2,y2) is given by
212
212 yyxxd
The Distance Formula
Distance Formula
MartinGay, Intermediate Algebra, 5ed 4040
Find the distance between (5, 8) and (2, 2).
212
212 yyxxd
22 28)2(5 d
22 63 d
5345369 d
The Distance Formula
Example:
MartinGay, Intermediate Algebra, 5ed 4141
The midpoint of the line segment whose endpoints are (x1,y1) and (x2,y2) is the point with coordinates
1 2 1 2,2 2
x x y y
The Midpoint Formula
Midpoint Formula
MartinGay, Intermediate Algebra, 5ed 4242
Find the midpoint of the line segment that joins points P(5, 8) and P(2, 2).
1 2 1 2,2 2
x x y y
The Midpoint Formula
5 2( ) ( ) ,82
22
7 10,2 2
3.5,5
Example:
MartinGay, Intermediate Algebra, 5ed 4343
§ 7.4
Adding, Subtracting, and Multiplying Radical
Expressions
MartinGay, Intermediate Algebra, 5ed 4444
Sums and Differences
Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient.
We can NOT split sums or differences.
baba
baba
MartinGay, Intermediate Algebra, 5ed 4545
In previous chapters, we’ve discussed the concept of “like” terms.
These are terms with the same variables raised to the same powers.
They can be combined through addition and subtraction.
Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand.Like radicals are radicals with the same index and the same radicand.
Like radicals can also be combined with addition or subtraction by using the distributive property.
Like Radicals
MartinGay, Intermediate Algebra, 5ed 4646
373 38
24210 26
3 2 42 Can not simplify
35 Can not simplify
Adding and Subtracting Radical Expressions
Example:
MartinGay, Intermediate Algebra, 5ed 4747
Simplify the following radical expression. 331275
3334325
3334325
333235
3325 36
Adding and Subtracting Radical Expressions
Example:
MartinGay, Intermediate Algebra, 5ed 4848
Simplify the following radical expression.
91464 33
9144 3 3 145
Adding and Subtracting Radical Expressions
Example:
MartinGay, Intermediate Algebra, 5ed 4949
Simplify the following radical expression. Assume that variables represent positive real numbers.
xxx 5453 3 xxxx 5593 2
xxxx 5593 2
xxxx 5533
xxxx 559
xxx 59 xx 510
Adding and Subtracting Radical Expressions
Example:
MartinGay, Intermediate Algebra, 5ed 5050
nnn abba
0 if b b
a
b
an
n
n
n a n bIf and are real numbers,
Multiplying and Dividing Radical Expressions
MartinGay, Intermediate Algebra, 5ed 5151
Simplify the following radical expressions.
xy 53 xy15
23
67
ba
ba
23
67
ba
ba44ba 22ba
Multiplying and Dividing Radical Expressions
Example:
MartinGay, Intermediate Algebra, 5ed 5252
§ 7.5
Rationalizing Numerators and Denominators of Radical Expressions
MartinGay, Intermediate Algebra, 5ed 5353
Many times it is helpful to rewrite a radical quotient with the radical confined to ONLY the numerator.
If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator.
This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the denominator.
Rationalizing the Denominator
MartinGay, Intermediate Algebra, 5ed 5454
Rationalize the denominator.
2
3
2
2
3 9
6
3
3
3
3
22
23
2
6
33
3
39
3 6
3
3
27
3 6
3
3 6 33 3 2
Rationalizing the Denominator
Example:
MartinGay, Intermediate Algebra, 5ed 5555
Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical.
In that case, we need to multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing).
The conjugate uses the same terms, but the opposite operation (+ or ).
Conjugates
MartinGay, Intermediate Algebra, 5ed 5656
Rationalize the denominator.
32
23
332322
3222323
32
32
32
322236
1
322236
322236
Rationalizing the Denominator
Example:
MartinGay, Intermediate Algebra, 5ed 5757
An expression rewritten with no radical in the numerator is called rationalizing the numerator.
7
12
7
12
12
12
212
12
127
1212 84
12
214
12
214
12
21
6
Rationalizing the Numerator
Example:
MartinGay, Intermediate Algebra, 5ed 5858
Rationalize the numerator.
3
7
9y3
3
7
9y
3 2
3 2
3
3
y
y
3 23
3 23
37
39
y
yy
3 2
3 3
21
27
y
y3 221
3
y
y
yx
yx
yx
yx
yxyxyx
yxyxyx
yyxyxx
yyxyxx
yxyx
yx
2
Rationalizing the Numerator
Example:
MartinGay, Intermediate Algebra, 5ed 6060
The Power Rule
Power RuleIf both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions.
A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.
MartinGay, Intermediate Algebra, 5ed 6161
Solving a Radical Equation1) Isolate one radical on one side of the equation.
2) Raise each side of the equation to a power equal to the index of the radical and simplify.
3) If the equation still contains a radical term, repeat Steps 1 and 2. If not, solve the equation.
4) Check all proposed solutions in the original equation.
Solving Radical Equations
MartinGay, Intermediate Algebra, 5ed 6262
Solve the following radical equation.
011 x
11 x
2211 x
11x
2x
2 1 1 0 011
011 true
Substitute into the original equation.
So the solution is x = 2.
Solving Radical Equations
Example:
MartinGay, Intermediate Algebra, 5ed 6363
Solve the following radical equation.
812 xx
xx 281
22281 xx
2432641 xxx 2433630 xx )421)(3(0 xx
213 or 4
x
Solving Radical Equations
Example:
MartinGay, Intermediate Algebra, 5ed 6464
Substitute the value for x into the original equation, to check the solution.
3 32( ) 1 8 846 true
21 21 14
24
8
84
25
2
21
82
5
2
21
82
26 falseSo the solution is x = 3.
Example continued:
Solving Radical Equations
MartinGay, Intermediate Algebra, 5ed 6565
Solve the following radical equation.425 yy
22425 yy
44445 yyy
445 y
44
5 y
22
44
5
y
416
25y
16
89
16
254 y
Solving Radical Equations
Example:
MartinGay, Intermediate Algebra, 5ed 6666
Substitute the value for x into the original equation, to check the solution.
5 289 891
46 16
16
252
16
169
4
52
4
13
4
3
4
13 false So the solution is .
Solving Radical Equations
Example continued:
MartinGay, Intermediate Algebra, 5ed 6767
Solve the following radical equation.24342 xx
43242 xx
2243242 xx
43434442 xxx
4343842 xxx
43412 xx
22 43412 xx6448)43(16144242 xxxx
080242 xx
0420 xx
20or 4x
Solving Radical Equations
Example:
MartinGay, Intermediate Algebra, 5ed 6868
Substitute the value for x into the original equation, to check the solution.
2( ) 4 3( 4 24 4) 2164
242
true
2( ) 4 3( ) 420 20 2
26436
286
true
So the solution is x = 4 or 20.
Solving Radical Equations
Example continued:
MartinGay, Intermediate Algebra, 5ed 6969
Solve the following radical equation.
51 x
2251 x
251x
24x
24 1 5
525 true
Substitute into the original equation.
So the solution is x = 24.
Solving Radical Equations
Example:
MartinGay, Intermediate Algebra, 5ed 7070
Solve the following radical equation.
55 x
2255 x
255 x
5x
5 55 525
Does NOT check, since the left side of the equation is asking for the principal square root.
So the solution is .
Substitute into the original equation.
Solving Radical Equations
Example:
MartinGay, Intermediate Algebra, 5ed 7171
Pythagorean Theorem
If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then
a2 + b2 = c2
There are several applications in this section that require the use of the Pythagorean Theorem in order to solve.
The Pythagorean Theorem
MartinGay, Intermediate Algebra, 5ed 7272
Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches.
c2 = 22 + 72 = 4 + 49 = 53
53c = inches
Using the Pythagorean Theorem
Example:
MartinGay, Intermediate Algebra, 5ed 7474
Imaginary Numbers
Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.
Imaginary UnitThe imaginary unit i, is the number whose square is – 1. That is,
2 1 and 1i i
MartinGay, Intermediate Algebra, 5ed 7575
Write the following with the i notation.
25 125
32 132 1216
121 1121
5 i
11 i
24 i 24 i
The Imaginary Unit, i
Example:
MartinGay, Intermediate Algebra, 5ed 7676
Real numbers and imaginary numbers are both subsets of a new set of numbers.
Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers.
Complex Numbers
MartinGay, Intermediate Algebra, 5ed 7777
Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers.
a is a real number and bi would be an imaginary number.
If b = 0, a + bi is a real number.
If a = 0, a + bi is an imaginary number.
Standard Form of Complex Numbers
MartinGay, Intermediate Algebra, 5ed 7878
Write each of the following in the form of a complex number in standard form a + bi.
6 = 6 + 0i
8i = 0 + 8i
24 164 62 i
256 1256 6 + 5i
620 i
Standard Form of Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 7979
Sum or Difference of Complex Numbers
If a + bi and c + di are complex numbers, then their sum is
(a + bi) + (c + di) = (a + c) + (b + d)i
Their difference is
(a + bi) – (c + di) = (a – c) + (b – d)i
Adding and Subtracting Complex Numbers
MartinGay, Intermediate Algebra, 5ed 8080
Add or subtract the following complex numbers. Write the answer in standard form a + bi.
(4 + 3) + (6 – 2)i = 7 + 4i
(8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i
(4 + 6i) + (3 – 2i) =
Adding and Subtracting Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 8181
The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms.
Multiplying Complex Numbers
MartinGay, Intermediate Algebra, 5ed 8282
Note that the product rule for radicals does NOT apply for imaginary numbers.
2516 ii 54 220i )1(20 20
2516 2516 20400
Multiplying Complex Numbers
MartinGay, Intermediate Algebra, 5ed 8383
Multiply the following complex numbers.
8i · 7i
56i2
56(1)
56
Multiplying Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 8484
Multiply the following complex numbers. Write the answer in standard form a + bi.
5i(4 – 7i)
20i – 35i2
20i – 35(–1)
20i + 35
35 + 20i
Multiplying Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 8585
Multiply the following complex numbers. Write the answer in standard form a + bi.
(6 – 3i)(7 + 4i)
42 + 24i – 21i – 12i2
42 + 3i – 12(–1)
42 + 3i + 12
54 + 3i
Multiplying Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 8686
Complex Conjugates
The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and
(a + bi)(a – bi) = a2 + b2
Complex Conjugate
MartinGay, Intermediate Algebra, 5ed 8787
The conjugate of a + bi is a – bi.
The conjugate of a – bi is a + bi.
The product of (a + bi) and (a – bi) is
(a + bi)(a – bi)
a2 – abi + abi – b2i2
a2 – b2(–1)
a2 + b2, which is a real number.
Complex Conjugate
MartinGay, Intermediate Algebra, 5ed 8888
Use complex conjugates to divide the following complex numbers. Write the answer in standard form.
i
i
34
26
i
i
i
i
34
34
34
26
2
2
9121216
681824
iii
iii
)1(916
)1(62624 i
25
2618 ii
25
26
25
18
Dividing Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 8989
Divide the following complex numbers.
i6
5
i
i
i 6
6
6
5
236
30
i
i
)1(36
30i
36
30ii
6
5
Dividing Complex Numbers
Example:
MartinGay, Intermediate Algebra, 5ed 9090
1i
12 i
iiiii )1(23
1)1)(1(224 iii
iiiii )1(45
1)1)(1(246 iii
iiiii ))(1(347
1)1)(1(448 iii
The powers recycle through each multiple of 4.
14 ki
Patterns of i