Download Math 80 Chapter 6 Lecture Notes

Math 80
Chapter 6
Lecture Notes
Professor Miguel Ornelas
1
M. Ornelas
Math 80 Lecture Notes
Section 6.1
Section 6.1
Rational Exponents
Principal and Negative Square Roots
If a is a nonnegative number, then
√
a is the principal square root and
√
− a is the negative square root of a.
Simplify. Assume that all variables represent positive numbers.
a.
d.
√
49
√
b.
√
√
0.64
e.
√
g. − 36
h.
r
0
c.
√
z8
√
−36
16
81
f.
16x4
i.
√
−2 9x12
Cube Root
√
The cube root of a real number a is written as 3 a.
a.
d.
√
3
−1
√
3
x12
b.
e.
√
3
r
27
p
3
c.
−8x6
f.
Definition
√
If n is a positive integer greater than 1 and n a is a real number, then
√
a1/n = n a
Section 6.1 continued on next page. . .
2
3
p
4
8
125
81x24 y 16
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Math 80 Lecture Notes
Section 6.1 (continued)
Use radical notation to write the following. Simplify if possible.
a.
361/2
d. −251/2
b.
641/3
c.
(−25)1/2
e.
(125x9 )1/3
f.
(81x8 )1/4
Write the radical with a rational exponent then simplify.
a.
p
3
x6 y 12
b.
√
4
81r8 s20
Definition
If m and√n are positive integers greater than 1, then
√ am/n = n am = n a
Simplify as much as possible.
a.
43/2
b.
82/3
c.
−163/4
d.
(−27)2/3
e.
16−3/4
f.
(−64)−2/3
Section 6.1 continued on next page. . .
3
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Math 80 Lecture Notes
Simplify as much as possible.
a. x3/4 · x5/4
c.
x7/8
x3/4
b.
(y 2/3 )3/4
d.
(27a3 b6 )1/3
(81a8 b−4 )1/4
b.
p
Section 6.2
Simplifying Radical Expressions
Product
Rule
for Radicals
√
√
√
n
n
n
a · b = ab
Quotient
Rule for Radicals
r
√
n
a
a
n
= √
n
b
b
Write the expressions in simplified form.
a.
c.
√
50
√
3
40a5 b4
Section 6.2 continued on next page. . .
d.
4
√
3
48x4 y 3
54a6 b2 c4
Section 6.1 (continued)
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Math 80 Lecture Notes
Section 6.2 (continued)
Rationalizing the Denominator
Simplify.
r
r
a.
3
2
b.
5
6
d.
√
2 3x
√
5y
e.
3
√
3
4
f.
(−3)2
b.
p
3
(−5)3
c.
25x2
e.
p
4
(x − 2)4
f.
c.
4
√
3
r
3
2
9
√
Finding n an = |a|
√
If n is even, then n an = |a|
√
If n is odd, then n an = a
Simplify.
a.
p
√
d.
√
5
x4
p
5
(2x − 7)5
M. Ornelas
Math 80 Lecture Notes
Section 6.3
Section 6.3
Adding and Subtracting Radical Expressions
Examples
Add or subtract as indicated. Assume all variables represent positive real numbers.
a.
a.
c.
√
√
3 17 + 5 17
√
√
b.
√
√
3
3
6 5z − 12 5z
√
20 + 2 45
b.
√
√
27x − 2 9x + 72x
d.
√
3
c.
√
√
3
3
54 − 5 16 + 2
p
3
48y 4 +
p
3
6y 4
Section 6.4
Multiplying and Dividing Radical Expressions
Examples
Multiply.
a.
√
3(5 +
√
30)
Section 6.4 continued on next page. . .
b.
6
√
√ √
( 2 − 5)( 6 − 2)
√
√
√
3
3 2+5 2−9 2
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c.
Math 80 Lecture Notes
√
( 6 − 3)2
√
( x + 2 + 3)2
d.
Rationalizing Denominators
Rationalize the denominator of each expression.
a.
2
√
5
r
c.
e.
3
1
2
2
3 2+4
√
b.
√
2 16
√
9x
d.
√
f.
√
6+2
√
6−2
7
5
3+2
Section 6.4 (continued)
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Math 80 Lecture Notes
Section 6.5
Section 6.5
Radical Equations
Examples
Solve.
a.
c.
e.
√
√
√
2x − 3 = 9
b.
−10x − 1 + 3x = 0
d.
4−x=x−2
f.
Section 6.5 continued on next page. . .
8
√
4x + 1 + 3 = 2
√
3
√
4
x+1+5
5x − 8 =
√
4
4x − 1
M. Ornelas
Math 80 Lecture Notes
Section 6.5 (continued)
Section 6.6
Complex Numbers
Imaginary Unit
The imaginary unit, written i, is the number whose square is −1. That is,
√
i = −1 and i2 = −1.
Write with i notation.
√
−49
a.
b.
Multiply or divide as indicated.
√
√
a.
−5 · −6
c.
√
125 ·
√
−5
√
√
c. − −20
−7
b.
d.
√
−16 ·
√
−1
√
−27
√
3
Complex Number
A complex number is a number that can be written in the form a + bi, where a and b are real numbers.
Add or subtract the complex numbers.
a. (3 − 5i) + (−8 + i)
b.
Section 6.6 continued on next page. . .
9
7i − (1 + 3i)
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Multiply the complex numbers.
a. 5i(3 + 2i)
Math 80 Lecture Notes
b.
(2 + 3i)(5 − i)
Section 6.6 (continued)
c.
(6 + 7i)(6 − 7i)
Complex Conjugates
The complex numbers (a + bi)(a − bi) are called complex conjugates of each other, and
(a + bi)(a − bi) = a2 + b2
Divide.
a.
5
3i
b.
4−i
2+i
Finding Powers of i
Find the following powers of i.
a. i9
b. i37
c.
10
i−56