Download Chapter 8 Rational Exponents, Radicals, and

Chapter 8
Rational
Exponents,
Radicals, and
Complex
Numbers
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CHAPTER
8
Rational Exponents, Radicals,
and Complex Numbers
8.1
8.2
8.3
8.4
8.5
8.6
8.7
Radical Expressions and Functions
Rational Exponents
Multiplying, Dividing, and Simplifying
Radicals
Adding, Subtracting, and Multiplying
Radical Expressions
Rationalizing Numerators and
Denominators of Radical Expressions
Radical Equations and Problem Solving
Complex Numbers
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8.7
Complex Numbers
1. Write imaginary numbers using i.
2. Perform arithmetic operations with complex
numbers.
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Imaginary unit: The number represented by i, where
i  1 and i2 = 1.
Imaginary number: A number that can be expressed
in the form bi, where b is a real number and i is the
imaginary unit.
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Example
Write each imaginary number as a product of a real
number and i.
a. 16
b. 21
c. 32
Solution
a. 16
b.
21
c.
32
 116
 1 21
 1 32
 1  16
 i4
 4i
 1  21
 1  32
 i 21
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 i 16  2
 4i 2
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Rewriting Imaginary Numbers
To write an imaginary number  n in terms of the
imaginary unit i,
1. Separate the radical into two factors, 1  n .
2. Replace 1 with i.
3. Simplify n .
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Complex number: A number that can be expressed in
the form a + bi, where a and b are real numbers and i
is the imaginary unit.
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Example
Add or subtract. (9 + 6i) + (6 – 13i)
Solution
We add complex numbers just like we add
polynomials—by combining like terms.
(9 + 6i) + (6 – 13i) = (9 + 6) + (6i – 13i )
= –3 – 7i
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Example
Add or subtract. (3 + 4i) – (4 – 12i)
Solution
We subtract complex numbers just like we subtract
polynomials.
(3 + 4i) – (4 – 12i) = (3 + 4i) + (4 + 12i)
= 7 + 16i
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Example
Multiply.
a. (8i)(4i)
b. (6i)(3 – 2i)
Solution
a. (8i)(4i)  32i 2
b. (6i)(3 – 2i)
2

18
i

12
i
 32(1)
 18i  12(1)
 32
 18i 12
 12 18i
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continued
Multiply.
c. (9 – 4i)(3 + i)
d. (7 – 2i)(7 + 2i)
Solution
c. (9 – 4i)(3 + i)
 27  9i  12i  4i 2
d. (7 – 2i)(7 + 2i)
 49  14i  14i  4i 2
 27  3i  4(1)
 49  4( 1)
 27  3i  4
 49  4
 31  3i
 53
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Complex conjugate: The complex conjugate of a
complex number a + bi is a – bi.
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Example
7
Divide. Write in standard form. 3i
Solution Rationalize the denominator.
7
7 i
7i
 

3i
3i i
3
7i
 2
3i
7i

3(1)
7i

3
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Example
3  5i
Divide. Write in standard form.
5i
Solution Rationalize the denominator.
15  3i  25i  5
3  5i 3  5i 5  i



25  1
5i
5i 5i
10  28i
2

15  3i  25i  5i

26
2
25  i
10 28i
15  3i  25i  5(1)



26 26
25  (1)
5 14i
 
13 13
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Example
40
Simplify. a. i
b. i33
Solution
a. i
40
= i

4 10
= 110 = 1
Write i40 as (i4)10.
b. i 33 = i 32  i
= i

4 8
= 1 i
i
Write i32 as (i4)8.
Replace i4 with 1.
=i
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