Post on 22-Dec-2015
40
Examples:
4 10
If and are real numbers, then a ba b a b Product Rule for Square Roots
2 10
7 75 7 25 3 7 5 3 35 3
6.2 – Simplified Form for Radicals
1716x xx1616 xx84
3 1716x 3 21528 xx 3 25 22 xx
104
3257
16
81
Examples:
2
5
4
9
45
49
aIf and are real numbers and 0, then
b
aa b b
b
Quotient Rule for Square Roots
2
25
9 5
7
3 5
7
16
81
2
25
45
49
6.2 – Simplified Form for Radicals
15
3
90
2
aIf and are real numbers and 0, then
b
aa b b
b
3 5
3
3 5
3
5
9 10
2
9 2 5
2
9 2 5
2
3 5
6.2 – Simplified Form for Radicals
Rationalizing the DenominatorRadical expressions, at times, are easier to work with if the denominator does not contain a radical. The process to clear the denominator of all radicals is referred to as rationalizing the denominator
5 3
5
x
5 31
5
x
5 53
5 5
x
5 15
25
x 5 15
5
x
15x
6.2 – Simplified Form for Radicals
5
3
7
20
5 3
3 3 5 3
3
7 20
20 20
7 20
20
7 4 5
20
2 35
20 35
10
7
20
7
4 5
7
2 5
7 5
2 5 5
35
2 25
35
2 5
35
10
6.2 – Simplified Form for RadicalsExamples:
2
45x
2
45x
2
9 5 x
2
3 5x
2 5
3 5 5
x
x x
10
3 5
x
x
10
15
x
x
6.2 – Simplified Form for RadicalsExamples:
Examples:
Theorem:6.2 – Simplified Form for Radicals
If “a” is a real number, then .
√ 40𝑥2
√ 4 ∙1 0 𝑥2
2|𝑥|√1 0
√𝑥2−16 𝑥+64
√ (𝑥− 8 )2
|𝑥− 8|
√18 𝑥3 − 9𝑥2
√9 𝑥2 (2 𝑥−1 )
3|𝑥|√2 𝑥−1
5 3x x
Review and Examples:
6 11 9 11
8x
15 11
12 7y y 5y
7 3 7 2 7
6.3 - Addition and Subtraction of Radical Expressions
27 75
Simplifying Radicals Prior to Adding or Subtracting
3 20 7 45
9 3 25 3
3 4 5 7 9 5
3 3 5 3 8 3
3 2 5 7 3 5
6 5 21 5 15 5
36 48 4 3 9 6 16 3 4 3 3
6 4 3 4 3 3 3 8 3
6.3 - Addition and Subtraction of Radical Expressions
Simplifying Radicals Prior to Adding or Subtracting
6 63 310 81 24p p
6 63 310 27 3 8 3p p
2 23 310 3 3 2 3p p
2 328 3p
2 23 330 3 2 3p p
6.3 - Addition and Subtraction of Radical Expressions
4 3 39 36x x x
Simplifying Radicals Prior to Adding or Subtracting
2 2 23 6x x x x x
23 6x x x x x
23 5x x x
6.3 - Addition and Subtraction of Radical Expressions
11x
Examples:
77
25
y
8
27
x
67
25
y y
3 7
5
y y
10x x 5x x
418x 49 2x 23 2x
8
9 3
x
4
3 3
x8
27
x
6.3 - Addition and Subtraction of Radical Expressions
3 88
Examples:
381
8
310
27
3
3
81
8
3 27 3
2
3 8 11 32 11
3 10
3
3
3
10
27
33 3
2
3 3 727m n 3 3 63 m n n 2 33mn n
6.3 - Addition and Subtraction of Radical Expressions
5 2
7 7
10 2x x
If and are real numbers, then a ba b a b
10
49 7
6 3 18 9 2 3 2
220x 24 5x 2 5x
6.4 –Multiplication and Division of Radical Expressions
Examples:
7 7 3 7 7 7 3 49 21
5 3 5x x
5 3x x
7 21
25 3 25x x 5 3 5x x
5 15x x
2 3 5 15x x x
𝑥−√3 𝑥+√5 𝑥−√15
6.4 –Multiplication and Division of Radical Expressions
Examples:
Review:
(x + 3)(x – 3) x2 – 3x + 3x – 9 x2 – 9
(√𝑥+3 ) (√𝑥− 3 )
√𝑥2−3 √𝑥+3√𝑥−9𝑥− 9
6.4 –Multiplication and Division of Radical Expressions
3 6 3 6
2
5 4x
9 6 3 6 3 36 3 36 33
5 4 5 4x x
225 4 5 4 5 16x x x
5 8 5 16x x
6.4 –Multiplication and Division of Radical Expressions
Examples:
2 5
2 1
22 5
2 11
1
2
4 2 5 2 5
4 2 2 1
2 6 2 5
2 1
7 6 2
1
7 6 2
If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required in order to rationalize the denominator.
conjugate
6.4 –Multiplication and Division of Radical Expressions
3
2 7
7
2 77
2
2
3
6 3 7
4 2 7 2 7 49
6 3 7
4 7
6 3 7
3
3 2 7
3
2 7 2 7
6.4 –Multiplication and Division of Radical Expressions
Example:
7
2 x
22
27 x
xx
2
7 2
4 2 2
x
x x x
7 2
4
x
x
6.4 –Multiplication and Division of Radical ExpressionsExample:
Radical Equations:
2 7x 6 1x x 9 2x
The Squaring Property of Equality:2 2, .If a b then a b
2 26, 6 .If x then x
2 25 2, 5 2 .If x y then x y
Examples:
6.5 – Equations Involving Radicals
Suggested Guidelines:
1) Isolate the radical to one side of the equation.
2) Square both sides of the equation.
3) Simplify both sides of the equation.
4) Solve for the variable.
5) Check all solutions in the original equation.
6.5 – Equations Involving Radicals
1
5x
6 1x x
2 2
6 1x x
6 1x x
5 1 0x
5 1x
1 16 1
5 5
6 11
5 5
6 5 1
5 5 5
1 1
5 5
6.5 – Equations Involving Radicals
52323 x
3323 x
333 332 x
2732 x
242 x
12x
5231223
523243
52273
523
55
6.5 – Equations Involving Radicals
3215 xx
115 xx
22115 xx
115 xxxx
1215 xxx
xx 224
01544 2 xx
01414 xx
01x
1x
014 x
22 224 xx
xxx 441616 2
042016 2 xx
4
1x
6.5 – Equations Involving Radicals
321115
32115
314
312
33
1x4
1x
324114
15
322114
5
323
41
323
21
32
6.5 – Equations Involving Radicals
1 5x x
1 5x x
2 2
1 5x x 21 10 25x x x
20 11 24x x
0 3 8x x 3 0 8 0x x
3 8x x
3 1 3 5
8 1 8 5
4 3 5 2 3 5
1 5
9 8 5
3 8 5 5 5
6.5 – Equations Involving Radicals
6.6 – Complex Numbers
1i
25 251 251 25i
Complex Number System:
This system of numbers consists of the set of real numbers and the set of imaginary numbers.
Imaginary Unit:
The imaginary unit is called i, where and .12 i
Square roots of a negative number can be written in terms of i.
i5
3 3i
32 32i 216i 24i
31
321
1i
72 72 ii 142i 14
The imaginary unit is called i, where and
.12 i
Operations with Imaginary Numbers
2
82
8i
2
24i
2
22i
125 ii 25 25i 5
327 327 i 81i i9
i2
6.6 – Complex Numbers
1iThe imaginary unit is called i, where and .12 i
Complex Numbers:
dicbia dibica idbca
idbca
Numbers that can written in the form a + bi, where a and b are real numbers.
3 + 5i 8 – 9i –13 + i
The Sum or Difference of Complex Numbers
dicbia cicbia dibica
6.6 – Complex Numbers
ii 35 215i )( 115
ii 643
ii 262 2412 ii )1(412 i
2424318 iii )( 142118 i
Multiplying Complex Numbers
i124
i2122
6.6 – Complex Numbers
15
42118 i
ii 5656 225303036 iii )( 12536
Multiplying Complex Numbers
61
221 i 24221 iii
441 i i43
ii 2121
6.6 – Complex Numbers
)( 1441 i
2536
i3
ii
i
32
3ii
13i
Dividing Complex NumbersRationalizing the Denominator:
6.6 – Complex Numbers
ii
64
ii
ii
64
2
2
64iii
)( 1614
i
614
i
61
64
i
i3
32
61 i
i
i
32
3
i
i
i
i
32
32
32
3
2
2
9664
3296
iii
iii
94
3116
i
13
113 ii
13
11
13
3
Dividing Complex NumbersComplex Conjugates:The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,
(a + bi)(a – bi) = a2 + b2
6.6 – Complex Numbers
i
i
76
94
i
i
i
i
76
76
76
94
2
2
49424236
63542824
iii
iii
4936
638224
i
85
8239 ii
85
82
85
39
Dividing Complex NumbersComplex Conjugates:The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,
(a + bi)(a – bi) = a2 + b2
6.6 – Complex Numbers
i5
6i
i
i 5
5
5
6
225
30
i
i
25
30ii
5
6
Dividing Complex NumbersComplex Conjugates:The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,
(a + bi)(a – bi) = a2 + b2
6.6 – Complex Numbers