Slide 1 / 272 Slide 2 / 272
Algebra II
Radical Equations
www.njctl.org
2015-04-21
Slide 3 / 272
Roots and RadicalsTable of Contents:
Working with Square Roots
Cube Rootsnth Roots
Irrational Roots
Solving Radical EquationsRational Exponents
Rationalizing the Denominator
Adding and Subtracting Radicals
Graphing Square Root Functions
Complex Numbers
Multiplying Radicals
click on the topic to go to that section
If review is needed before or during this unit click on the link below.
Fundamental Skills of Algebra (Supplemental Review)
Click for Link
Slide 4 / 272
Graphing Square Root Functions
Return to Table of Contents
Slide 5 / 272
The inverse of is
Recall the Inverse of Squares...
...but the result is not a function.
Slide 6 / 272
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, # )
Slide 7 / 272 Slide 8 / 272
3-3
The function is one of the parent functions. Use this fact to help you anticipate the graph or find the function from the graph.
Remember transformations...
Domain: [0, # )Range: [0, # ) Domain: [-3, # )
Range: [0, # )
Domain: [3, # )Range: [0, # )
Parent
Slide 9 / 272
-3
3
And...
Domain: [0, # )Range: [0, # )
Domain: [0, # )Range: [3, # )
Domain: [0, # )Range: [-3, # )
Parent
Slide 10 / 272
1 Which is the graph of the function?
A B
C D
Teac
her
Slide 11 / 272
2 Which is the graph of the function?
A B
C D
Teac
her
Slide 12 / 272
3
3 Which is the equation of the graph?
A
B
C
DTe
ache
r
Slide 13 / 272 Slide 14 / 272
Slide 15 / 272 Slide 16 / 272
Slide 17 / 272Remember what happens when you have af(x) or f(bx)...
Domain: [0, # )Range: [0, # )
Domain: [0, # )Range: (-# , 0]
Domain: [0, # )Range: [0, # )
Parent
Slide 18 / 272
And...
Domain: (-# , 0]Range: (-# , 0]
Domain: (-# , 0]Range: [0, # )
Domain: [0, # )Range: [0, # )
Parent
Slide 19 / 272
8 Which is the graph of the function?A B
C D
Teac
her
Slide 20 / 272
9 Which is the graph of the function?A B
C D
Teac
her
Slide 21 / 272
10 Which is the equation of the graph?
A
B
C
D
Teac
her
Slide 22 / 272
11 Which is the equation of the graph?
A
B
C
D
Teac
her
Slide 23 / 272
What happens when you combine the transformations? Why is this one only moved to the right 2?
Teac
her
Slide 24 / 272
Slide 25 / 272
In order to see how much the graph moves horizontally, any coefficient must be factored out. Rewrite the following by factoring out the coefficient:
Teac
her
Slide 26 / 272
Slide 27 / 272 Slide 28 / 272
Slide 29 / 272 Slide 30 / 272
Slide 31 / 272 Slide 32 / 272Graph the function:
Teac
her
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.
Slide 33 / 272
Graph the function:Te
ache
r1. 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.
Slide 34 / 272
Graph the function:
Teac
her
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.
Slide 35 / 272 Slide 36 / 272
Working with Square Roots
Return to Table
of Contents
Slide 37 / 272 Slide 38 / 272
Slide 39 / 272
17 What is ?Te
ache
r
Slide 40 / 272
18 Find:
Teac
her
Slide 41 / 272
19 What is ?
Teac
her
Slide 42 / 272
20 What is ?
Teac
her
Slide 43 / 272 Slide 44 / 272
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.
Variables
Slide 45 / 272
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.
Slide 46 / 272
Slide 47 / 272 Slide 48 / 272
Slide 49 / 272 Slide 50 / 272
Slide 51 / 272 Slide 52 / 272
27
A
B
C
D no real solution
Teac
her
Slide 53 / 272
28
A
B
C
D no real solution
Teac
her
Slide 54 / 272
29
A
B
C
D no real solutionTe
ache
r
Slide 55 / 272
30
A
B
C
D no real solution
Teac
her
Slide 56 / 272
Return to Table
of Contents
Irrational Roots
Slide 57 / 272
Simplifying Radicals
is said to be a rational number because there is a perfect square that equals the radicand.
If a radicand cannot be made into a perfect square, the root is said to be irrational, like .
Slide 58 / 272
The commonly accepted form of a radical is called simplest radical form.
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.
Slide 59 / 272 Slide 60 / 272
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)Te
ache
r
Slide 61 / 272
32 Which of the following is the prime factorization of 72?
A 9(8)
B 2(2)(2)(2)(6)
C 2(2)(2)(3)
D 2(2)(2)(3)(3)
Teac
her
Slide 62 / 272
33 Which of the following is the prime factorization of 12?
A 3(4)
B 2(6)
C 2(2)(2)(3)
D 2(2)(3)
Teac
her
Slide 63 / 272
34 Which of the following is the prime factorization of 24 rewritten as powers of factors?
A
B
C
D
Teac
her
Slide 64 / 272
35 Which of the following is the prime factorization of 72 rewritten as powers of factors?
A
B
C
D
Teac
her
Slide 65 / 272 Slide 66 / 272
36 Simplify:
A
B
C
D already in simplified formTe
ache
r
Slide 67 / 272
37 Put in simplest radical form:
A
B
C
D already in simplified form
Teac
her
Slide 68 / 272
38 Put in simplest radical form:
A
B
C
D already in simplified form
Teac
her
Slide 69 / 272
39 Simplify:
A
B
C
D already in simplified form
Teac
her
Slide 70 / 272
40 Which of the following is not an irrational number?
A
B
C
D
Teac
her
Slide 71 / 272 Slide 72 / 272
Slide 73 / 272
42 Simplify:
A
B
C
D
Teac
her
Slide 74 / 272
Slide 75 / 272
44 Put in simplest radical form:
A
B
C
D
Teac
her
Slide 76 / 272
Slide 77 / 272
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.
Slide 78 / 272
Examples:
Teac
her
Slide 79 / 272
46 Simplify:
A
B
C
D
Teac
her
Slide 80 / 272
47 Put in simplest radical form:
A
B
C
D
Teac
her
Slide 81 / 272
48 Simplify:
A
B
C
D
Teac
her
Slide 82 / 272
49 Put in simplest radical form:
A
B
C
D
Teac
her
Slide 83 / 272
50 Put in simplest radical form:
A
B
C
D
Teac
her
Slide 84 / 272
Adding and Subtracting Radicals
Return to Table of Contents
Slide 85 / 272
Adding and Subtracting Radicals*Note: When adding or subtracting radicals, you do not add or subtract the radicands (the inside).
Consider:
Slide 86 / 272
To add and subtract radicals they must be like terms.
Radicals are like terms if they have the same radicands and the same indexes.
Like Terms Unlike Terms
Slide 87 / 272 Slide 88 / 272
51 Identify all of the pairs of like terms:A
B
C
D
E
F
Teac
her
Slide 89 / 272 Slide 90 / 272
Slide 91 / 272 Slide 92 / 272
Slide 93 / 272
53 Simplify:
A
B
C
D Already Simplified
Teac
her
Slide 94 / 272
Slide 95 / 272 Slide 96 / 272
Slide 97 / 272 Slide 98 / 272
Slide 99 / 272
57 Simplify:
A
B
C
D Already in simplest form
Teac
her
Slide 100 / 272
Slide 101 / 272 Slide 102 / 272
60 Simplify:
A
B
C
D Already in simplest formTe
ache
r
Slide 103 / 272 Slide 104 / 272
Slide 105 / 272
Multiplying Radicals
Return to Table
of Contents
Slide 106 / 272
Slide 107 / 272
Whole number times whole number and radical times radical. Never multiply a whole number and radical! Leave
all answers in simplest radical form.
Teac
her
Slide 108 / 272
Examples:
Teac
her
Slide 109 / 272
Examples:
Teac
her
Slide 110 / 272
63 Multiply:
A
B
C
D
Teac
her
Slide 111 / 272
64 Simplify:
A
B
C
D
Teac
her
Slide 112 / 272
65 Simplify:
A
B
C
D
Teac
her
Slide 113 / 272
66 Simplify:
A
B
C
D
Teac
her
Slide 114 / 272
67 Simplify:
A
B
C
DTe
ache
r
Slide 115 / 272
Multiplying Polynomials with RadicalsLeave all answers in simplest radical form
Teac
her
Slide 116 / 272
68 Multiply and write in simplest form:
A
B
C
D
Teac
her
Slide 117 / 272
69 Multiply and write in simplest form:
A
B
C
D
Teac
her
Slide 118 / 272
70 Multiply and write in simplest form:
A
B
C
D
Teac
her
Slide 119 / 272
71 Multiply and write in simplest form:
A
B
C
D
Teac
her
Slide 120 / 272
72 Multiply and write in simplest form:
A
B
C
DTe
ache
r
Slide 121 / 272
Rationalizing the Denominator
Return to Table
of Contents
Slide 122 / 272
Mathematicians don't like radicals in the denominators of fractions.
When there is one, the denominator is said to be irrational. The method used to rid the denominator is termed "rationalizing the denominator".
Which of these has a rational denominator?
RationalDenominator
IrrationalDenominator
Rationalizing the Denominator
Teac
her
Slide 123 / 272
If the denominator is a monomial, to rationalize, just multiply top and bottom of the fraction by the root part of the denominator.
Examples:
Teac
her
Slide 124 / 272
Slide 125 / 272
Multiplying by the conjugate turns an irrational number into a rational number.
Check out what happens... Teac
her
Slide 126 / 272
Do you see a pattern that let's us go from line 1 to line 3 directly?Example Example Example
Teac
her
Slide 127 / 272 Slide 128 / 272
Use conjugates to rationalize the denominators:
Teac
her
Slide 129 / 272
73 What is conjugate of ?
A
B
C
D
Teac
her
Slide 130 / 272
Slide 131 / 272
75 Simplify:
A
B
C
D Already simplified
Teac
her
Slide 132 / 272
76 Simplify:
A
B
C
D Already simplified
Teac
her
Slide 133 / 272
77 Simplify:
A
B
C
D Already simplified
Teac
her
Slide 134 / 272
Slide 135 / 272 Slide 136 / 272
80 Simplify:
A
B
C
D
Teac
her
Already simplified
Slide 137 / 272 Slide 138 / 272
Cube Roots
Return to Table
of Contents
Slide 139 / 272
If a square root cancels a square, what cancels a cube?
Teac
her
Slide 140 / 272
Slide 141 / 272 Slide 142 / 272
Try...
Teac
her
Slide 143 / 272 Slide 144 / 272
Slide 145 / 272 Slide 146 / 272
Slide 147 / 272 Slide 148 / 272
Slide 149 / 272 Slide 150 / 272
Slide 151 / 272
Put in simplest radical form:
Teac
her
Slide 152 / 272
87 Simplify:
A B
C D not possible
Teac
her
Slide 153 / 272
88 Simplify:
A B
C D not possible
Teac
her
Slide 154 / 272
89 Simplify:
A B
C D not possible
Teac
her
Slide 155 / 272
90 Simplify:
A B
C D not possible
Teac
her
Slide 156 / 272
91 Simplify:
A B
C D not possibleTe
ache
r
Slide 157 / 272
92 Put in simplest radical form:
A
B
C
D not possible
Teac
her
Slide 158 / 272
nth Roots
Return to Table
of Contents
Slide 159 / 272 Slide 160 / 272
Try...
Teac
her
Slide 161 / 272 Slide 162 / 272
Slide 163 / 272
95 Simplify:
A
B
C
D
Teac
her
Slide 164 / 272
96 Simplify:
A
B
C
D
Teac
her
Slide 165 / 272 Slide 166 / 272
98 Simplify:
A
B
C
D
Teac
her
Slide 167 / 272
99 Simplify:
A
B
C
D
Teac
her
Slide 168 / 272
100 Simplify:
A
B
C
DTe
ache
r
Slide 169 / 272 Slide 170 / 272
Slide 171 / 272
Try...Te
ache
r
Slide 172 / 272
Slide 173 / 272 Slide 174 / 272
103 Simplify:
A
B
C
D
Teac
her
Slide 175 / 272
104 Simplify:
A
B
C
D
Teac
her
Slide 176 / 272
Slide 177 / 272
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:
Rationalizing nth roots of monomialsTe
ache
r
Slide 178 / 272
Try:
Teac
her
Slide 179 / 272
106 Rationalize:
A
B
C
D
Teac
her
Slide 180 / 272
Slide 181 / 272
108 Rationalize:
A
B
CD
Teac
her
Slide 182 / 272
Slide 183 / 272 Slide 184 / 272
Slide 185 / 272
112 Simplify:
A
B
C
D
Teac
her
Slide 186 / 272
Rational Exponents
Return to Table
of Contents
Slide 187 / 272
Rational exponents, or exponents that are fractions, are another way to write and work with radicals.
Power
Root
Slide 188 / 272
Slide 189 / 272
Simplify:Te
ache
r
Slide 190 / 272
Slide 191 / 272 Slide 192 / 272
115 Simplify:
Teac
her
Slide 193 / 272
116 Simplify:
Teac
her
Slide 194 / 272
117 Simplify:
Teac
her
Slide 195 / 272 Slide 196 / 272
Rewrite each radical as a rational exponent in the lowest terms.
Teac
her
Slide 197 / 272
Rewrite each expression as a single radical. To combine more than one number or variable, the roots must be the same. Te
ache
r
Slide 198 / 272
When the roots (denominators) are different, they must be made into a common number in order to create a single root.
Teac
her
Slide 199 / 272 Slide 200 / 272
Slide 201 / 272
120 Find the simplified expression that is equivalent to:
A
B
C
D
Teac
her
Slide 202 / 272
121 Find the simplified expression that is equivalent to:
A
B
C
D
Teac
her
Slide 203 / 272
122 Find the simplified expression that is equivalent to:
A
B
C
D
Teac
her
Slide 204 / 272
123 Simplify:
A
B
C
D
Teac
her
Slide 205 / 272
124 Write with rational exponents:
A
B
C
D
Teac
her
Slide 206 / 272
125 Find the simplified expression that is equivalent to:
A
B
C
D
Teac
her
Slide 207 / 272
126 Write the following with exponents:
A
B
C
D
Teac
her
Slide 208 / 272
Slide 209 / 272
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. Te
ache
r
Slide 210 / 272
Slide 211 / 272
128 Simplify:
A
B
C
D
Teac
her
Slide 212 / 272
Slide 213 / 272
130 Simplify:
A
B
C
D
Teac
her
Slide 214 / 272
Slide 215 / 272 Slide 216 / 272
Solving RadicalEquations
Return to Table
of Contents
Slide 217 / 272 Slide 218 / 272
Slide 219 / 272
Example:Te
ache
r
Slide 220 / 272
Slide 221 / 272
133 Find the solution to:
Teac
her
Slide 222 / 272
134 Find the solution to:
Teac
her
Slide 223 / 272
135 Find the solution to:
Teac
her
Slide 224 / 272
Slide 225 / 272
137 Find the solution to:Te
ache
r
Slide 226 / 272
Slide 227 / 272
138 Solve the following:
Teac
her
Slide 228 / 272
Slide 229 / 272
140 Solve:
Teac
her
Slide 230 / 272
Slide 231 / 272
Complex Numbers
Return to Table of Contents
Slide 232 / 272
Slide 233 / 272
Rational Numbers
Irrational Numbers
Imaginary Numbers
Real Numbers
Whole Numbers
Natural Numbers
Complex Numbers: All numbers are technically considered complex numbers. Real Numbers can be written as a + 0i - no imaginary component.
Integers
Slide 234 / 272
Complex Numbers
Why does this work?
Teac
her
Slide 235 / 272
Higher order i's can be simplified into i, -1, -i, or 1.
If the power of i is even: If the power of i is odd:...and the exponent is a multiple of 4, then it simplifies to 1.
...and the exponent is a multiple of 2,but not 4, then it simplifies to -1.
...factor out one i to create an even exponent. Use the rules for even exponents and leave the factored i.
Slide 236 / 272
Slide 237 / 272 Slide 238 / 272
142 Simplify:
A i
B -1
C -i
D 1
Teac
her
Slide 239 / 272
143 Simplify:
A i
B -1
C -i
D 1
Teac
her
Slide 240 / 272
144 Simplify:
A i
B -1
C -i
D 1Te
ache
r
Slide 241 / 272
145 Simplify:
A i
B -1
C -i
D 1
Teac
her
Slide 242 / 272
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.
Teac
her
Slide 243 / 272 Slide 244 / 272
Slide 245 / 272
147 Simplify:
A
B
C
D
Teac
her
Slide 246 / 272
Slide 247 / 272 Slide 248 / 272
150 Simplify:
A
B
C
D
Teac
her
Slide 249 / 272
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.
Slide 250 / 272
Slide 251 / 272 Slide 252 / 272
When multiplying, multiply numbers, multiply i's and simplify any i with a power greater than one.
Teac
her
Slide 253 / 272 Slide 254 / 272
Multiply and leave answers in standard form.
Teac
her
Slide 255 / 272 Slide 256 / 272
151 Simplify:
A
B
C
D
Teac
her
Slide 257 / 272
152 Simplify:
A
B
C
D
Teac
her
Slide 258 / 272
153 Simplify:
A
B
C
DTe
ache
r
Slide 259 / 272
154 Simplify:
A
B
C
D
Teac
her
Slide 260 / 272
155 Simplify:
A
B
C
D
Teac
her
Slide 261 / 272
Dividing with iSince 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.
Teac
her
Slide 262 / 272
Slide 263 / 272 Slide 264 / 272
156 Simplify:
A B
C DTe
ache
r
Slide 265 / 272
157 Simplify:
A B
C D
Teac
her
Slide 266 / 272
Slide 267 / 272 Slide 268 / 272
Slide 269 / 272 Slide 270 / 272
160 Simplify:
A
B
C
DTe
ache
r
Slide 271 / 272
161 Simplify:
A
B
C
D
Teac
her
Slide 272 / 272
162 Simplify:
A
B
C
D
Teac
her