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Algebra II
Radical Equations
2015-04-21
www.njctl.org
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Roots and Radicals
Table of Contents:
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click on the topic to go
to that section
Graphing Square Root Functions
Working with Square Roots
Irrational Roots
Adding and Subtracting Radicals
Multiplying Radicals
Rationalizing the Denominator
Cube Roots
nth Roots
Rational Exponents
Solving Radical Equations
Complex Numbers
Graphing Square Root
Functions
Return to
Table of
Contents
If review is needed before or during this unit click on the link below.
Fundamental Skills of Algebra
(Supplemental Review)
Click for Link
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Recall the Inverse of Squares...
The inverse of
is
...but the result is not a function.
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The domain of y = x2 is restricted to x ≥ 0, so that the inverse will be a
function.
Domain: [0, # )
Range: [0, # )
Domain: [0, # )
Range: [0, # )
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The function
is one of theparent functions. Use this fact to
help you anticipate the graph or find the function from the graph.
Remember transformations...
Parent
Domain: [0, # )
Range: [0, # )
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3
A
B
C
D
Teacher
1 Which is the graph of the function?
-3
Domain: [0, # )
Range: [3, # )
Domain: [0, # )
Range: [-3, # )
2 Which is the graph of the function?
B
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Teacher
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3 Which is the equation of the graph?
Teacher
Domain: [0, # )
Range: [0, # )
A
Domain: [3, # )
Range: [0, # )
Domain: [-3, # )
Range: [0, # )
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And...
Parent
3
-3
A
B
C
D
C
D
3
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Remember what happens when you have af(x) or f(bx)...
Parent
And...
Parent
Domain: [0, # )
Range: [0, # )
Domain: [0, # )
Range: [0, # )
Domain: [0, # )
Range: (-# , 0]
Domain: [0, # )
Range: [0, # )
Domain: (-# , 0]
Range: [0, # )
Domain: (-# , 0]
Range: (-# , 0]
C
B
9 Which is the graph of the function?
A
B
C
D
D
A
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Teacher
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10 Which is the equation of the graph?
11 Which is the equation of the graph?
A
B
B
C
C
D
D
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Teacher
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What happens when you
combine the transformations?
Why is this one only moved to
the right 2?
Teacher
A
Teacher
8 Which is the graph of the function?
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Teacher
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In order to see how much the graph moves horizontally, any
coefficient must be factored out. Rewrite the following by
factoring out the coefficient:
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Teacher
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Teacher
Graph the function:
1. Anticipate the graph using
transformations.
2. Set the radicand ≥ 0 and solve to find
the domain.
3. Choose 3 values in the domain that
will give you perfect squares under the
radical. Use a t-table to find y's.
4. Plot the points and graph.
1. Anticipate the graph using
transformations.
2. Set the radicand ≥ 0 and solve to find
the domain.
3. Choose 3 values in the domain that
will give you perfect squares under the
radical. Use a t-table to find y's.
4. Plot the points and graph.
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Graph the function:
Teacher
Graph the function:
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Teacher
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1. Anticipate the graph using
transformations.
2. Set the radicand ≥ 0 and solve to find
the domain.
3. Choose 3 values in the domain that
will give you perfect squares under the
radical. Use a t-table to find y's.
4. Plot the points and graph.
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Working with Square Roots
Return to
Table
of Contents
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18
Find:
Teacher
?
19 What is
?
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Teacher
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20 What is
?
Teacher
What is
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Teacher
17
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Variables
What happens when you have variables in the
radicand? To take the square root of a variable rewrite
its exponent as the square of a power.
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IMPORTANT: When taking the square root of variables, remember that
answers must be positive. Even powered answers, like the last page, will
be positive even if the variables are negative. The same cannot be said if
the answer has an odd power. When you take a square root and the
answer has an odd power, put the result inside an absolute value
symbol.
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Teacher
27
A
C
B
D no real solution
28
Teacher
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Teacher
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29
A
C
A
C
B
D no real solution
B
D no real solution
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Teacher
30
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A
C
B
D no real solution
Irrational Roots
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Table
of Contents
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Simplifying Radicals
If a radicand cannot be made into a perfect square, the
root is said to be irrational, like
.
To simplify numbers that are not perfect squares, start by breaking the
radicand into factors and then breaking the factors into factors and so
on until only prime numbers are left. This is called prime factorization.
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31
Which of the following is the prime factorization of 24?
A
3(8)
B
4(6)
C
2(2)(2)(3)
D
2(2)(2)(3)(3)
Teacher
is said to be a rational number because there is a perfect
square that equals the radicand.
The commonly accepted form of a radical is called
simplest radical form.
Which of the following is the prime factorization of 12?
A
3(4)
2(2)(2)(2)(6)
B
2(6)
C
2(2)(2)(3)
C
2(2)(2)(3)
D
2(2)(2)(3)(3)
D
2(2)(3)
A
9(8)
B
34
Which of the following is the prime factorization of 24
rewritten as powers of factors?
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Teacher
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Teacher
33
35
Which of the following is the prime factorization of 72
rewritten as powers of factors?
A
A
B
B
C
C
D
D
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Teacher
Which of the following is the prime factorization of 72?
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36
Simplify:
A
B
C
D already in simplified form
Teacher
32
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Teacher
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38
Put in simplest radical form:
A
A
B
B
C
C
D already in simplified form
D already in simplified form
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Simplify:
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Teacher
39
A
B
40
Which of the following is not an irrational number?
A
B
C
D already in simplified form
C
D
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Teacher
Put in simplest radical form:
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Teacher
37
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Teacher
42 Simplify:
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A
B
C
D
44 Put in simplest radical form:
A
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B
C
D
The same process goes for variables, but absolute value
signs need to be included where appropriate.
Absolute value symbols are required when the initial exponent is
even and the exponent after taking the root is odd. If the initial
exponent is odd, you will not need absolute values.
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Examples:
Teacher
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47 Put in simplest radical form:
A
A
B
B
C
C
D
D
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49 Put in simplest radical form:
A
A
B
B
C
C
D
D
50 Put in simplest radical form:
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Teacher
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Teacher
Teacher
48 Simplify:
Teacher
Teacher
46 Simplify:
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A
B
C
D
Adding and
Subtracting Radicals
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Table of
Contents
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To add and subtract radicals they must be like terms.
Adding and Subtracting Radicals
Radicals are like terms if they have the same radicands
and the same indexes.
*Note: When adding or subtracting radicals, you do not add or
subtract the radicands (the inside).
Like Terms
Consider:
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51
Identify all of the pairs of like terms:
A
B
C
D
E
F
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Teacher
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Unlike Terms
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Simplify:
Teacher
53
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A
B
C
D
Already Simplified
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Teacher
Simplify:
A
B
C
D Already in simplest form
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60
Simplify:
A
B
C
D Already in simplest form
Teacher
57
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Multiplying Radicals
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of Contents
Examples:
Teacher
Whole number times whole number and radical times
radical. Never multiply a whole number and radical! Leave
all answers in simplest radical form.
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Teacher
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Examples:
63
Multiply:
Teacher
Teacher
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A
B
C
D
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65
Simplify:
A
A
B
B
C
C
D
D
66
Simplify:
A
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Teacher
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67
Simplify:
A
B
B
C
C
D
Teacher
Teacher
Simplify:
D
Teacher
64
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68
Multiply and write in simplest form:
Teacher
Multiplying Polynomials with Radicals
Leave all answers in simplest radical form
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Teacher
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A
B
C
D
70
Multiply and write in simplest form:
A
A
B
B
C
C
D
D
71
Multiply and write in simplest form:
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Teacher
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Teacher
Multiply and write in simplest form:
72
Multiply and write in simplest form:
A
A
B
B
C
C
D
D
Teacher
69
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Rationalizing the Denominator
Mathematicians don't like radicals in the denominators of fractions.
Teacher
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When there is one, the denominator is said to be irrational. The
method used to rid the denominator is termed "rationalizing the
denominator".
Rationalizing the
Denominator
Which of these has a rational denominator?
Rational
Denominator
Irrational
Denominator
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Table
of Contents
If the denominator is a monomial, to rationalize, just multiply
top and bottom of the fraction by the root part of the
denominator.
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Teacher
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Examples:
Check out what happens...
Do you see a pattern that let's us go from line 1 to line 3 directly?
Example
Example
Example
Teacher
Multiplying by the conjugate turns an irrational number into
a rational number.
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Teacher
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Use conjugates to rationalize the denominators:
73
What is conjugate of
?
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Teacher
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A
B
C
D
Simplify:
A
B
C
D Already simplified
76
Simplify:
A
B
C
D Already simplified
Teacher
75
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Teacher
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Simplify:
Teacher
A
B
C
D Already simplified
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80 Simplify:
Teacher
77
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A
B
C
D
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Already simplified
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Cube Roots
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of Contents
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Teacher
If a square root cancels a square, what cancels a cube?
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Teacher
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Try...
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87 Simplify:
A
B
C
D not possible
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Teacher
88 Simplify:
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Teacher
Put in simplest radical form:
Teacher
Teacher
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89 Simplify:
A
B
A
B
C
D not possible
C
D not possible
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91 Simplify:
Teacher
Teacher
90 Simplify:
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A
B
A
B
C
D not possible
C
D not possible
92 Put in simplest radical form:
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Teacher
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A
B
nth Roots
C
not possible
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Table
of Contents
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Try...
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Teacher
D
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96
Simplify:
A
A
B
B
C
C
D
D
Teacher
Simplify:
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Simplify:
A
C
B
D
Simplify:
A
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Teacher
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99
Teacher
98
100
Simplify:
A
B
B
C
C
D
D
Teacher
95
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Teacher
Try...
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103
Simplify:
A
B
C
D
Teacher
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104
Teacher
Simplify:
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A
B
C
D
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Rationalizing nth roots of monomials
Teacher
Try:
Teacher
Remember that
, given an nth root in the denominator, it will need to
be rationalized. To rationalize, find the complement if the nth root that will
create a perfect root in the denominator. Multiply top and bottom by the
complement. Simplify.
Examples:
106
Rationalize:
A
B
C
D
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Teacher
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108
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Teacher
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Rationalize:
A
B
C
D
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Teacher
112 Simplify:
A
C
B
D
Rational Exponents
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of Contents
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Rational exponents, or exponents that are fractions, are
another way to write and work with radicals.
Power
Root
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Teacher
Simplify:
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115 Simplify:
Teacher
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117 Simplify:
Teacher
Teacher
116 Simplify:
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Teacher
Rewrite each radical as a rational exponent in the lowest terms.
When the roots (denominators) are different, they must be made
into a common number in order to create a single root.
Teacher
Rewrite each expression as a single radical. To combine
more than one number or variable, the roots must be the
same.
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Teacher
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Find the simplified expression that is equivalent to:
121
A
Find the simplified expression that is equivalent to:
A
B
B
C
C
D
D
A
123 Simplify:
Teacher
Find the simplified expression that is equivalent to:
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A
B
B
C
C
D
D
Teacher
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122
Teacher
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Teacher
120
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125
Find the simplified expression that is equivalent to:
A
A
B
B
C
C
Teacher
124 Write with rational exponents:
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D
D
126 Write the following with exponents:
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A
B
C
D
Just like other problems where you must rationalize denominators,
mathematicians like to have a an integer power in the denominators.
Therefore, if there is a fractional exponent in the denominator after
simplifying, rationalize the denominator.
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Teacher
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128
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Teacher
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Simplify:
A
B
C
D
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130
A
B
Teacher
Simplify:
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C
D
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Solving Radical
Equations
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of Contents
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Teacher
Example:
Find the solution to:
134
Find the solution to:
Teacher
133
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135
Find the solution to:
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Teacher
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137
Find the solution to:
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Teacher
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138
Solve the following:
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Teacher
140 Solve:
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Complex Numbers
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of Contents
Complex Numbers: All numbers are technically considered complex numbers. Real Numbers
can be written as a + 0i - no imaginary component.
Real Numbers
Rational Numbers
be
ag
Im
ina
ry
m
Nu
rs
Integers
Whole Numbers
Natural
Numbers
Irrational Numbers
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Complex Numbers
Why does this work?
Teacher
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Higher order i's can be simplified into i, -1, -i, or 1.
...factor out one i to
create an even exponent.
Use the rules for even
exponents and leave the
factored i.
...and the exponent is a
multiple of 2,but not 4,
then it simplifies to -1.
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142
Simplify:
A
i
B
-1
C
-i
D
1
143
Simplify:
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Teacher
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Teacher
If the power of i is odd:
...and the exponent is a
multiple of 4, then it
simplifies to 1.
144
Simplify:
A i
A
i
B -1
B
-1
C -i
C
-i
D 1
D
1
Teacher
If the power of i is even:
A
i
B
-1
C
-i
D
1
147
Simplify:
A
B
C
D
Teacher
Simplify:
Simplify radical expressions that have a negative by taking out i first.
Then, perform the indicated operation(s). Simplify any expression
that has a power of i greater than one.
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Teacher
145
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150
Teacher
Simplify:
A
B
C
D
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Working with Complex Numbers
Operations, such as addition, subtraction, multiplication and
division, can be done with i.
Treat i like any other variable, except at the end make sure i is
at most to the first power.
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When multiplying, multiply numbers, multiplyi's and simplify any i
with a power greater than one.
Teacher
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Teacher
Multiply and leave answers in standard form.
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151
Teacher
Simplify:
A
B
C
D
Simplify:
153
Simplify:
A
A
B
B
C
C
D
D
Teacher
152
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Teacher
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155
Simplify:
A
A
B
B
C
C
D
D
Dividing with i
Since i represents a square root, a fraction is not in simplified form if
there is an i in the denominator. And, similar to roots, if the
denominator is a monomial just multiply top and bottom of the fraction
by i to rationalize.
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Teacher
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156
Simplify:
Teacher
154 Simplify:
Teacher
Teacher
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A
B
C
D
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Simplify:
Teacher
A
B
C
D
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160
Teacher
157
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Simplify:
A
C
B
D
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Teacher
Simplify:
A
C
B
D
162
Simplify:
Teacher
161
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A
C
B
D