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 3
5
x
5 31
5
x
5 53
5 5
x
5 15
25
x 5 15
5
x
15x
7.5 – Rationalizing the Denominator of Radicals Expressions
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
7.5 – Rationalizing the Denominator of Radicals Expressions
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
7.5 – Rationalizing the Denominator of Radicals Expressions
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
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
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
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
7.5 – Rationalizing the Denominator of Radicals Expressions
conjugate
7
2 x
22
27 x
xx
2
7 2
4 2 2
x
x x x
7 2
4
x
x
conjugate
7.5 – Rationalizing the Denominator of Radicals Expressions
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
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
2 7x
2 2
2 7x
2 49x
51x
51 2 7
49 7
7 7
7.6 – Radical Equations and Problem Solving
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
7.6 – Radical Equations and Problem Solving
9 2x
7x
2 2
7x
49x
49 9 2
7 9 2
16 2
no solution
7.6 – Radical Equations and Problem Solving
7.6 – Radical Equations and Problem Solving
52323 x
3323 x
333 332 x
2732 x
242 x
12x
5231223
523243
52273
523
55
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
4
1x
7.6 – Radical Equations and Problem Solving
321115
32115
314
312
33
1x4
1x
324114
15
324114
5
347
41
347
21
34
9
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
7.6 – Radical Equations and Problem Solving
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
7.7 – Complex Numbers
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
7.7 – 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
7.7 – Complex Numbers
ii 3425 ii 3245 i9
342 i
ii 26 ii 26 i72
342 i i41
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
7.7 – Complex Numbers
ii 5656 225303036 iii 2536
Multiplying Complex Numbers
61
221 i 24221 iii
441 i i43
ii 2121
7.7 – Complex Numbers
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 Numbers
Complex Conjugates:
The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,
(a + bi)(a – bi) = a2 + b2
7.7 – 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 Numbers
Complex Conjugates:
The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,
(a + bi)(a – bi) = a2 + b2
7.7 – Complex Numbers
i5
6i
i
i 5
5
5
6
225
30
i
i
25
30ii
5
6
Dividing Complex Numbers
Complex Conjugates:
The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,
(a + bi)(a – bi) = a2 + b2