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Chapter 7
Powers, Roots and
Complex Numbers
• How are operations performed on
radical expressions?
Activation
•
•
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Radical Expressions
7-1
What is the definition of a square root?
Lesson
• Square root—
– The square root of a number “a” is
a number “c” such that c2 = a
–
root/index
radical
radicand
– Note:
• The radicand may not be negative
when the root is even
• k a k | a | when k is even
Examples
3
25
4
81
5
27
4
16
81
 32
Multiplying and
Simplifying 7-2
How do you multiply and simplify radical
expressions?
Lesson
•
Multiplying –
– Two radicals can be multiplied if
their roots/indexes are the same
– Ex. x  4 x  4  x 2 16
•
Simplifying –
– Just as two radicals can be
multiplied the numbers can be
separated for simplification
– Ex.
20  4 5  2 5
Examples
32
3
80
3 4
12ab c
3x  12x  12
2
HOMEWORK
•PAGE(S): 295-296
•NUMBERS: 4-36 by 4’s (not 28)
•PAGE(S): 299
•NUMBERS: 4 – 32 by 4’s
Activation
•
•
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Operations with
Radical Expressions 7-3
How do you apply arithmetic operations to
radical expressions?
Lesson
•
Dividing radicals—
– If the radicals do not have a
perfect root, then if they have the
same index divide the radicals
before simplifying
16
25
3
54
3
2
3
3
96a 4b 4
12a 3b
Lesson
•
Adding and Subtracting with radicals
– To add or subtract radicals you
must have the SAME index and
VALUE under the radical
94 5  34 5
53 16  33 54
HOMEWORK
•PAGE(S): 303
•NUMBERS: 4 – 48 by 4’s
Activation
•
•
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More Operations
with Radical
Expressions 7-4
How do you rationalize the denominator of
a radical expression?
Lesson
•
Multiplying radicals—
– Must have the same index to
multiply the radicands
( 4 3  2 )( 3  5 2 )
Lesson
•
Rationalizing the denominator
– Removing radicals from the
denominator through multiplication
7
5
3
3
7
5
5
2x4
5 3y
3 2
4 5
Lesson
•
Rationalizing the numerator
– Removing radicals from the
numerator through multiplication
7
5
3 2
4 5
HOMEWORK
•PAGE(S): 308
•NUMBERS: 4-48 by 4’s & 56
Activation
•
•
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Rational Numbers
as Exponents 7-5
What is the relationship between radical
expressions and rational exponents?
Lesson
•
Converting between radical and rational notation:
n
•
x x
m
m
n
Examples rewrite the expression:
x
3
x
4
5
x
2
3
HOMEWORK
•PAGE(S): 315
•NUMBERS: 12 – 60 by 4’s
Activation
•
•
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Solving Radical
Equations 7-6
How do you solve equations with radical
expressions?
Lesson
•
Solving radical equations
– Isolate the radical where possible
– Raise the to the appropriate power
– Solve for x
– Check for extraneous solutions
Examples
•
Solve the following:
x3  6
3
6x  9  8  5
Examples
3x  1  2 x  6
20  x  8  9  x  11
HOMEWORK
•PAGE(S): 319 - 320
•NUMBERS: 4 – 36 by 4’s and 44
Activation
•
Find
 25
 17
Imaginary and Complex
Numbers 7-7
How do you express the square root of a
negative number?
Lesson
•
Imaginary numbers—
the value given to a negative radicand
--used so that certain equations have a solution
i  1
•
Try the following
 25
 17
 1 25
 1 17
5i
i 17
Lesson
i  1
i 2  1 1  1
i 3  i 2 1  1 i  i
i4  i2  i2  1
The “Numbrella”
Complex numbers
a+bi
Has a real and an imaginary component
Imaginary Numbers
Real Numbers
All the numbers represented by
each point on the number line
Rational Numbers
Can be expressed as a fraction
from
Irrational Numbers
i or bi
i  1
Can’t be expressed as a fraction
Non-terminating, non-repeating integers
Integers
Positive and negative numbers and zero
All “non-decimal” values
Whole Numbers
All positive integers or the counting numbers and zero
Natural Numbers
All positive integers or the counting numbers 1, 2, 3, . . .
Lesson
 5  6i
•
To add, subtract, multiply or divide with imaginary
numbers—use the i as a variable until the end then
substitute the appropriate values
•
Examples:
5  6
 7i  10i
( 7i  2)(3i  4)
2 i
3  4i
HOMEWORK
•PAGE(S): 323
•NUMBERS: 4 – 40 by 4’s
•PAGE(S): 329
•NUMBERS: 19 – 22
Complex Numbers
and Graphing 7-8
How do you graph a complex number?
Activation
•
Graph the following points:
•
(1, 2)
•
(-3, 5)
•
(-2, -3)
Lesson
•
Plot each of the following:
•
3 + 2i
•
-4 + 5i
•
-5 – 4i
Lesson
•
Absolute value of a complex number:
we know the absolute value of real number can be
thought of as its distance from the origin. The same is true
of a complex number
•
Plot: 3 + 2i
32  22
2
3
HOMEWORK
•PAGE(S): 325
•NUMBERS: 2 – 20 Even
Activation
• Is 3 a solution to x2 + 4x + 3 = 0?
Solutions of Equations
7-9
How do you solve an equation with a
complex number?
How do you find complex solutions of
equations?
Lesson
•
Determine if
1  i 7 is a solution to x 2  2 x  8  0
(1  7 ) 2  2(1  7 )  8  0
Lesson
•
Find the equation that has 4 + 3i and 4 – 3i as solutions
x= 4 + 3i and x = 4 – 3i
x – 4 – 3i = 0
x – 4 + 3i =0
(x – 4 – 3i )(x – 4 + 3i ) = 0
x2 – 4x + 3xi – 4x + 16 – 12i – 3xi + 12i – 9i2 = 0
x2 – 8x + 16 – 9(-1) = 0
x2 – 8x + 25 =0
Lesson
•
Solve
3 – 4i + 2xi = 3i – (1 – i)x
2xi +(1- i)x = 7i – 3
(2i + 1 – i)x = 7i – 3
(i + 1)x = 7i – 3
x = 7i  3
i 1
HOMEWORK
•PAGE(S): 333
•NUMBERS: 6 – 21 by 3’s
Review
•worksheet