Review and Examples:
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Transcript of Review and Examples:
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5 3x x
Review and Examples:
6 11 9 11
8x
15 11
12 7y y 5y
7 3 7 2 7
7.4 – Adding, Subtracting, Multiplying Radical Expressions
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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
7.4 – Adding, Subtracting, Multiplying Radical Expressions
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4 3 39 36x x x
Simplifying Radicals Prior to Adding or Subtracting
6 63 310 81 24p p
2 2 23 6x x x x x
23 6x x x x x 23 5x x x
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
7.4 – Adding, Subtracting, Multiplying Radical Expressions
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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
7.4 – Adding, Subtracting, Multiplying Radical Expressions
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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
7.4 – Adding, Subtracting, Multiplying Radical Expressions
𝑥−√3 𝑥+√5 𝑥−√15
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3 6 3 6
25 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
7.4 – Adding, Subtracting, Multiplying Radical Expressions
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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 radical is referred to as rationalizing the denominator
5 35
x
5 3 15
x
5 535 5
x
5 1525
x 5 15
5x
15x
7.5 – Rationalizing the Denominator of Radicals Expressions
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53
720
5 33 3
5 33
7 2020 20
7 20
20
7 4 5
20
2 35
20 35
10
720
7
4 5
7
2 5
7 52 5 5
35
2 25
352 5
3510
7.5 – Rationalizing the Denominator of Radicals Expressions
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245x
2
45x
29 5 x
23 5x
2 53 5 5
xx x
10
3 5xx
1015
xx
7.5 – Rationalizing the Denominator of Radicals Expressions
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7.5 – Rationalizing the Denominator of Radicals Expressions
If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required.
Review:
(x + 3)(x – 3) x2 – 3x + 3x – 9 x2 – 9
(x + 7)(x – 7) x2 – 7x + 7x – 49 x2 – 49
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2 52 1
22 52 11
12
4 2 5 2 54 2 2 1
2 6 2 52 1
7 6 2
1
7 6 2
7.5 – Rationalizing the Denominator of Radicals Expressions
If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required.
conjugate
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32 7
72 77
22
3
6 3 7
4 2 7 2 7 49
6 3 74 7
6 3 73
3 2 7
3
2 7 2 7
7.5 – Rationalizing the Denominator of Radicals Expressions
conjugate
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72 x
22
27 xxx
2
7 2
4 2 2
x
x x x
7 2
4
x
x
conjugate
7.5 – Rationalizing the Denominator of Radicals Expressions
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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:
7.6 – Radical Equations and Problem Solving
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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.
7.6 – Radical Equations and Problem Solving
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2 7x
2 22 7x
2 49x
51x
51 2 7
49 7
7 7
7.6 – Radical Equations and Problem Solving
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15
x
6 1x x
2 26 1x x
6 1x x
5 1 0x
5 1x
1 16 15 5
6 115 5
6 5 15 5 5
1 15 5
7.6 – Radical Equations and Problem Solving
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9 2x
7x
2 27x
49x
49 9 2
7 9 2
16 2
no solution
7.6 – Radical Equations and Problem Solving
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7.6 – Radical Equations and Problem Solving
52323 x
3323 x
333 332 x
2732 x
242 x
12x
5231223
523243
52273
523
55
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7.6 – Radical Equations and Problem Solving
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
41
x
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7.6 – Radical Equations and Problem Solving
321115
32115
314
312
33
1x41
x
324114
15
324114
5
347
41
347
21
349
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1 5x x
1 5x x
2 21 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
7.6 – Radical Equations and Problem Solving
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7.7 – 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
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7.7 – Complex Numbers1i
72 72 ii 142i 14
The imaginary unit is called i, where and
.12 i
Operations with Imaginary Numbers
28
28i
224i
222i
125 ii 25 25i 5
327 327 i 81i i9
i2
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7.7 – Complex Numbers1iThe 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
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7.7 – Complex Numbers
ii 3425 ii 3245 i9
342 i
ii 26 ii 26 i72
342 i i41
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7.7 – Complex Numbers
ii 35 215i 15
ii 643
ii 262 2412 ii 412 i
2424318 iii 42118 i
Multiplying Complex Numbers
i124
i2122
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7.7 – Complex Numbers
ii 5656 225303036 iii 2536
Multiplying Complex Numbers
61
221 i 24221 iii
441 i i43
ii 2121
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7.7 – Complex Numbers
ii32
3
ii
ii
3232
323
2
2
96643296iiiiii
94
3116
i
13113 i
i1311
133
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
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7.7 – Complex Numbers
ii
7694
ii
ii
7676
7694
2
2
4942423663542824
iiiiii
4936638224
i
858239 i
i8582
8539
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
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7.7 – Complex Numbers
i56
ii
i 55
56
22530
ii
2530i
i56
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
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