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Chapter 7
7.1 Radical Expression
Theorem 7.1
• Every positive real number has two real
number square roots
• 0 has just one square root: 0
• Negative numbers do not have real number
square roots
Principal Square Root
• The Principal Square Root of a non-negative real number
is its non-negative root.
• Thesymbol a represents the principal squareroot of a.
• The negative square root of a is written as;  a
• This symbol is a radical sign
• An expression written with a radical sign is a radical
expression
• The expression written under the radical sign is the
radicand
Radical Expression
Index
Radical
k
a
Radicand
Absolute Value
• For any real number a,
a  a
2
• The principal square root of a2 is the
absolute value of a.
k
a  a ; When k is even
k
a  a; When k is odd
k
k
HW 7.1
Pg 295-296 1-39 Odd, 40-53
Chapter 7
7.2 Multiplying and Simplifying
HW 7.2
Pg 299 1-47
Chapter 7
7.3 Operations with Radical
Expressions
Chapter 7
7.4 More Operations with Radical
Expressions
HW 7.3-4
Pg 303 1-52
Pg 308-309 1-65 Odd, 66-69
Chapter 7
7.5 Rational Numbers as Exponents
Theorem 7.6: For any nonnegative number a, any natural
number index k, and any integer m,
k
3
3
a 
m
 a
k
m
27 2  ?
 2
8 ?

2
3
6
6y

?
3
?
Definition: For any nonnegative number a and any natural
1
k
number index k, a means
k
a (the nonnegative kth root of a)
Note: When working with rational exponents, we will
assume that variables in the base are nonnegative
1
2
x  x
1
3
27  3 27  3
Definition: For any natural numbers m and k, and any
m
k
nonnegative number a, a means
2
3
k
27  27 
3
2
am

3
27

2
m
m

Definition: For any rational number k and any real number a, a k
means 1
a
m
k
(5 xy )

4
5

1
(5 xy )
4
5
HW #7.5
Pg 315-316 3-75 Every third
problem, 78-83
Chapter 7
7.6 Solving Radical Equations
Isolate the radical expression on one side of the equation.
Note: When solving a radical equation it is sometimes necessary
to isolate the radical and use the inverse operation more than once
2x  5 1  x  3
HW #7.6
Pg 319-320 3-39 every third
problem 41-52
Chapter 7
7.7 Imaginary and Complex Numbers
Solve: x2 + 1 = 0
x2 = -1
x   1
In the real number system negative
numbers do not have square roots,
therefore there is no real solution.
Mathematicians invented imaginary numbers so
negative numbers would have square roots
Thus the solution to the above equation would be
x  i
Definition:
The imaginary numbers consist of all numbers
bi, where b is a real number and i is the imaginary unit, with the
property that i2 = -1
i  1
i  1
i 63 
i 28 
2
i  i
3
i 53 
i 1
4
i121 
Pattern Recognition Using the information from above, write a general
statement about the standard form of in where n is a positive integer. Use
this statement to write i231in standard form.
When operating on imaginary numbers:
1. Always take the i out of the radical first
2. Treat i as a variable
3. Never write i with a power greater than 1
1. 47i  2  94i
3. 6i  2i  12
2. 5  2i  2 5
4. 3  6  3 2
Definition:
The complex numbers consist of all sums a + bi,
where a and b are real numbers and i is the imaginary unit. The
real part is a and the imaginary part is bi.
All real numbers are complex numbers
We assume that i behaves like a real number, that is it
obeys all the rules of real numbers
34. 3(2  5i)2  (2  i)(4)  5i
55  69i
HW #7.7
Pg 323 1-33 Odd 34-63
Chapter 7
7-8 Complex Numbers and Graphing
Remember a complex number has a real part and an
imaginary part. These are used to plot complex
numbers on a complex plane.
z  a  bi
Imaginary
Axis
absolute value of z
z  a  bi The
denoted by |z| is the
z

a
b
Real
Axis
distance from the origin to
the point (x, y).
z  a b
2
2
Plot the number in the complex plane and then find the absolut
value of the complex number.
Plot the number in the complex plane and then find the
absolute value of the complex number.
Adding and Subtracting Complex
numbers Graphically
Adding and Subtracting Complex
numbers Graphically
HW #7.8
Pg 325 1-23
Chapter 7
7-9 More about Complex Numbers
Equality of Complex Numbers:
a +bi = c + di if and only if a = c and b = d
Solve the following equation for x and y:
5x + 6i = 10 + 2yi
Equality of Complex Numbers:
a +bi = c + di if and only if a = c and b = d
Solve the following equations for x and y:
3x + yi = 5x + 1 + 2i
Multiply
Multiply
Multiply
Conjugate of a complex number
The conjugate of a + bi is a - bi
Theorem 7-8 The product of a nonzero complex number
a + bi and its conjugate a - bi is a2 + b2
Prove Theorem 7-8
Find the reciprocal of 3 + 4i.
HW #7.9
Pg 329 1-27 Odd, 28-41
Chapter 7
7-10 Solutions of Equations
A Solution to an Equation is the number that when
substituted for the variable makes the equations true.
Is 2 + i a solution of x2 – 4x + 5 = 0?
A Solution to an Equation is the number that when
substituted for the variable makes the equations true.
Is 1  i 7 a solution of x2 – 2x + 8 = 0?
Principle of zero products: If ab = 0 , then a = 0 or b = 0.
Find an equation having 1 + 2i and 1 – 2i as solutions
Principle of zero products: If ab = 0 , then a = 0 or b = 0.
Find an equation having 4 +3i and 4 – 3i as solutions
Solve the following equations:
1. 3ix + 4 – 5i = (1 + i)x + 2i
2. 3 – 4i + 2ix = 3i - (1 – i)x
3. 3x + 4ix = 1 + i
Theorem 7-9 Every polynomial with complex coefficients and a
degree of n (where n >1) can be factored into n linear factors.
Show that (x + i)(x – i) is a factorization of x2 + 1
Show that (x + 2i)(x – 2i) is a factorization of x2 + 4
Show that [x - (2 + 3i)][(x – (2 - 3i)] is a factorization of x2 – 4x + 13
Theorem 7-10 Every polynomial of degree n (n  1) with
complex coefficients has at least one solution and at most n
solutions in the complex number system.
Theorem 7-11 Every nonzero complex number has two square
roots. They are additive inverses of each other. 0 has one square
root.
Show that 1 + i is the square root of 2i. Find the other square root.
Show that -1 + i is the square root of -2i. Find the other square root.
Show that 3 - i is the square root of 8 - 6i. Find the other square root.
HW #7.10
Pg 333 1-31 Odd, 32