Download Ch. 9 :: Workbook/Homework

Math 154 ::
Elementary Algebra
Chapter 9 — Radical Expressions and Equations
Section 9.1 — Introduction to Square Roots — Part I
Section 9.2 — Simplifying Radical Expressions
Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions
Section 9.4 — Division of Radical Expressions
Section 9.5 — Simplifying Radical Expressions — Part II
Section 9.6 — Radical Equations
Answers
Chapter 9 — Radical Expressions and Equations
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Math 154 :: Elementary Algebra
Section 9.1
Chapter 9 — Radical Expressions and Equations
Introduction to Square Roots
Example:
Simplify. If the expression is not a real number, state so.
49
a) 
4
To simplify a square root of a perfect square, use the fact that:
a  b if and only if a  b2 .
If a is nonnegative, then
49
4


a2  a .
7
1  
2
If a is negative, then
2

1
7
2


a 2  a
7
2
Homework
1.
In your own words, define the square root of a number.
2.
In your own words, define the square of a number.
3.
Does the square root of a negative number have a real number value?
4.
What is the expression under the square root sign called?
Simplify. If the expression is not a real number, state so.
5.
36
6.
1
7.
144
8.
25
4
9.
 100
10.
0
11.
4
12.
196
13.
1
49
14.
 81
15.

9
64
16.
225
17.
169
18.
 400
19.
1
900
20.
16  4
21.
25  9
Section 9.1 — Introduction to Square Roots
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Math 154 :: Elementary Algebra
Chapter 9 — Radical Expressions and Equations
22. Simplify each of the following.
a)
 4 2
b)
16 2
c)
x2
d)
 4 2
e)
 9 2
f)
x2
g)
 4
h)
 9
i)
 x
if x is a nonnegative value.
if x is a negative value.
2
2
2
Section 9.1 — Introduction to Square Roots
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Math 154 :: Elementary Algebra
Section 9.2
Chapter 9 — Radical Expressions and Equations
Simplifying Radical Expressions — Part I
For all problems in the remainder of Chapter 9, assume that all variables are nonnegative.
Examples:
Simplify. Assume that all variables are nonnegative. If the expression is not a real number, state so.
a)
1400x10 y9
One way to simplify a radical expression is to factor the radicand into primes.
2  2  2  5  5  7  x10 y9
2  5 2  7  x10 y9
In this problem, there are a pair of 2s and a pair of 5s. Taking the square root of these
“pairs”, will result in the following:
This problem may also be viewed as 1000 14  x10 y9 .
Now work on the variables. To take the square root of a variable raised to a specific exponent, you may divide the
EXPONENT by 2.
2  5  x5  y 4 2  7  y
Now, simplify by multiply the “outside” together and the “inside” together.
10 x5 y 4 14 y
b)
20a 4 c 15a12c7
The easiest way to simplify a product of two square roots is to write the expression as the square root of a single product.
20a 4c 15a12c7
Before taking the square root or multiplying the constants, find the prime factorization of
each constant.
2  2  5  3  5  a4  a12  c  c7
Multiply the variables together by adding their exponents.
2  2  5  5  3  a16  c8
This problem may also be viewed as 100  3  a16  c8 .
2  5  a8  c4 3
Now, simplify by multiply the “outside” together.
10a8c4 3
Homework
1.
In your own words, describe how to simplify the square root of a non-perfect square.
2.
When you “pull factors out from a square root”, what operation is between those factors and the factors that remain under the
square root?
3.
If the radicand in a square root expression has a variable raised to an exponent, what is the short cut rule for simplifying the
square root for that variable?
4.
In your own words, describe how simplifying each of the following expressions are different: 16 and
x16 .
Section 9.2 — Simplifying Radical Expressions — Part I
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Math 154 :: Elementary Algebra
5.
Chapter 9 — Radical Expressions and Equations
Determine whether each statement is true or false.
a)
The square root of a positive perfect square is a rational number.
b) A rational number is always real.
c)
A real number is always rational.
d) An irrational number is not real.
e)
The square root of a positive non-perfect square is a real number.
f)
The square root of a positive non-perfect square is an irrational number.
g) The square root of a negative perfect square is a real number.
h) An irrational number is real.
i)
The square root of a negative non-perfect square is a real number.
6.
When multiplying two single-term square root expressions, what’s the “easiest” step to take first?
7.
Assuming all variables are nonnegative, what is the simplified answer for a problem that “squares a square root” or “square
roots a square”? In other words, what is the relationship between “squaring” and “square rooting”?
Simplify. Assume that all variables are nonnegative. If the expression is not a real number, state so.
8.
18
9.
20
10.
45
11.
200
12.
50
13.
162
14.
288
15.
112
16.
1575
17.
363
18.
432
19.
3 44
20.
5 12
21.
10 28
22.
7 54
23.
x3
24.
z11
25.
k9
26.
n400
27.
p 401
Section 9.2 — Simplifying Radical Expressions — Part I
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Math 154 :: Elementary Algebra
28.
5y 4
29.
8a8
30.
16m16
31.
98c12 d 9
32.
108y 25 z10
33.
192 p6 q9
34.
320a 20 c17 d 2
35.
648v 21u 32
36.
8mn8
37.
490x50 y 42 z
38.
150a150
39.
540c36 d 25
Chapter 9 — Radical Expressions and Equations
Simplify. Assume that all variables are nonnegative. Note that there is no addition or subtraction in any of the following problems.
40.
5 5
41.
16  16
42.
 8
43.
 11
44.
2 6
45.
15 12
46.
18 42
47.
1 22
48.
20 45
49.
35 14
50.
63 28
51.
3x
52.
5 y 2 10 y
53.
6a3 15a5
54.
27 z 5 6 z 4
55.
40m3n 15mn3
2
2
3 x3
Section 9.2 — Simplifying Radical Expressions — Part I
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Math 154 :: Elementary Algebra
56.
8xy 4 z 2 xy3
57.
10a3c 35ac5
58.
48 p6 q 24 p6 q7
59.
60.
61.

25w6

 11k 
 3a b 
Chapter 9 — Radical Expressions and Equations
2
2
5
9
2
62.

9 x8

4 x10
63.

2 y3

7z

64.

5m
3m

65.

4cd
2
2

2

2

2
2
2
4cd

2
Section 9.2 — Simplifying Radical Expressions — Part I
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Math 154 :: Elementary Algebra
Section 9.3
Chapter 9 — Radical Expressions and Equations
Addition, Subtraction, and Multiplication of Radical Expressions
Examples:
Simplify.
a)
45  2 45  7 5
To add square roots, the radicands must be “like”. Before combining like-radicand terms, simplify each term by taking the
square root.
335  2 335  7 5
This problem may also be viewed as
95 2 95 7 5 .
3 5  23 5  7 5
Remember, the factor that is “pulled” out is multiplied by what remains in the
square root.
3 56 57 5
Combine like-radicand terms. The radicand remains the same; only the constant
in front of the radical changes.
4 5
Multiply. All answers must be given in simplified form.
b)
8
3 6

32 6

To multiply a two term square root expression by another expression, distribute.
8
3 6

8 3 3
8 33
32 6

In this problem, use the FOIL method.
8 32 6
8 2 36
8 9
16 18
83
16  3 2
24
 48 2
12
 45 2
 6 3
 63
 18
3 2
2 66
 2 36
3 2
 62 6
You may leave the problem as written to simplify each square root or
rewrite as:
This may be written as: 8 9
16 9  2
 92
 2 36
26
12
Combine like-radicand terms only.
Homework
1.
Can you add any two square root expressions? In your own words, explain why or why not.
2.
In your own words, describe how to add two square root expressions.
Simplify.
3.
4 2 5 2
4.
8 11  3 11
5.
4 10  10
6.
85 8
7.
9 14  14
8.
12 5  3 6  5
Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions
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Math 154 :: Elementary Algebra
9.
12  4 12  7 2
10.
75  12
11.
20  45
12.
108  3
13.
100  44
14.
2 63  175
15.
5 192  3 80
16.
2 150  6
17.
 121  18
18.
225  4 27
19.
2 36  275
20.
5 162  34  3 8
Chapter 9 — Radical Expressions and Equations
Multiply. All answers must be given in simplified form.
22.
13
23.
2
24.



7 5
2 5 2
21.

10  6


11 9  2




25.
6 3 52
26.
4 8 3 12
27.
x x 6
28.
y y
29.
2a 3 a  a
30.
5 3 54
31.
14 6  3 14
32.
5 11 8  3 11
33.
2 7 6  10 6





y













34.
10 3 2  4 5
35.
4 6 9 14  21


Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions
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Math 154 :: Elementary Algebra


36.
x 8 x  13
37.
3 a 3a  3 a
38.
4z 4 z  4z





5c 5c  5 c
39.



40.
2 7n 2 7n  7n
41.
 6  2  3  5 
42.
 8  3 1  7 
43.
 4  6 1 0  3 
44.
1 1  2  5  2 
45.
1 
46.
1 0  3 2  4  2 5 
47.
1 2  3  5  8 3 
48.
8
6 2

49.
3
2 7
 9
50.
4 
51.
1  3 
52.
 x  2 x  3x  x 
53.
y  z
54.
n  4
55.
 3  2 5  3  2 5 
56.

57.
 w  8 w  w  8 w 
58.
9
59.
 3z  z
60.
14
 7 

10
Chapter 9 — Radical Expressions and Equations
14

67 2

24 7

2
2
2
3n
 2n 
62 7
a  4a

 9
10
3n

62 7
a  4a
 3z  z


10

In your own words, describe the similarities in the problems 55 through 59 above. In your own words, describe the patter
and what is common about all of the answers.
Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions
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Math 154 :: Elementary Algebra
Section 9.4
Chapter 9 — Radical Expressions and Equations
Division of Radical Expressions
Examples:
Simplify. Assume all variables are nonnegative. You may need to rationalize the denominator in order to give a simplified answer.
a)
240 x12 y 7
140 x3 y11
To simplify the division of two square root expressions, the easiest first step is to write the radicands under one square root
sign and then reduce or simplify the fraction.
240 x12 y 7
12 x9

140 x3 y11
7 y4
Next, take the square root of the numerator and the square root of the denominator.
4  3  x9

7  y4
2 x 4 3x
y2 7
If the above two steps result in an expression that does NOT have a radical in the denominator, the problem is simplified. If
there is still a radical in the denominator, rationalize the denominator.
For this problem, there is still a radical in the denominator, so rationalize the denominator by multiplying the fraction by
2 x 4 3x
y2 7

7
7

7
7
2 x 4 21x
7 y2
Rationalize the denominator.
2 3
b)
1 5
To rationalize the denominator of an expression that contains two terms in the denominator, multiply the numerator and
denominator by the conjugate of the denominator.
1  5 
5  1  5 
2 3
1 


2 3  1 5

1 5



2 3  1 5

4
Notice how the denominator is multiplied (FOIL-ed) out, but the numerator is left in factored form.
In many cases, this will be the last step, but for this problem, the fraction can be simplified as it contains a factor of 3 in the
numerator and denominator.

2 3 1 5
42




3 1 5
2
 Your answer can also be written as 
3  15
.
2
Homework
1.
When simplifying a square root whose radicand is a quotient, what is usually the “easiest” first step?
2.
When looking at an expression that consists of a quotient of square roots (without an addition or subtraction), should you take the
square root first or simplify the fraction first?
Section 9.4 — Division of Radical Expressions
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Math 154 :: Elementary Algebra
Chapter 9 — Radical Expressions and Equations
3.
What are the three conditions that must be met in order for a square root expression to be considered “simplified”?
4.
In your own words, describe, in general what it means to rationalize a denominator.
5.
What are the two different types of expressions you will see in this class where you might need to rationalize the denominator? In
your own words, describe how to rationalize the denominator for each type.
Simplify. Assume all variables are nonnegative.
25
6.
9
7.
75
3
8.
16
4
9.

18
2
10.
10
5
11.
24
3
12.
100
2
13.
18
72
14.
15.
16.
17.
18.
19.
20.
21.
45
5
105
21
98
18
110
90
56
7
x7
x3
40m8
125m6
28a8 c
63a 6 c
Section 9.4 — Division of Radical Expressions
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Math 154 :: Elementary Algebra
22.
23.
24.
25.
26.
27.
Chapter 9 — Radical Expressions and Equations
2 y4
162 y12
78wz10
24w5 z
75 p11q 21
27 p13 q5
80m25 n9
16m3 n 4
3x 4 y8 z
192 x6 y12 z
1815c16 d 25
15c9 d
Simplify. Assume all variables are nonnegative. You may need to rationalize the denominator in order to give a simplified answer.
8
28.
3
29.
7
2
30.
6
10
31.
12
15
32.
33.
34.
35.
36.
37.
38.
14
7
7
14
5
y
4a 2
20a 3
8
2p
x8
x11
m13
18m10
Section 9.4 — Division of Radical Expressions
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Math 154 :: Elementary Algebra
39.
40.
41.
42.
43.
44.
45.
Chapter 9 — Radical Expressions and Equations
52 z 4
26 z 2
180c15 d 10
640c 2 d 16
168 x13 y 3
84 x14 y10
810 pq 4
2430 p 2 q 6
33m17 n3
88mn6
63a 20 c
35a11c
120 xy 3
180 x 4 y8
Rationalize the denominator.
5
46.
2 3
47.
48.
49.
50.
51.
52.
53.
54.
55.
1
3 7
2
1 6
8
3 5
4 10
3 2
6 3
8 5
x
8 x
11
11  y
2 a
a c
xy
x y
Section 9.4 — Division of Radical Expressions
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Math 154 :: Elementary Algebra
Section 9.5
Chapter 9 — Radical Expressions and Equations
Simplifying Radical Expressions — Part II
9.5 — Simplifying Radical Expressions — Part II Worksheet
Example:
Simplify.
6  36  24
a)
4
To simplify an expression like the one above, follow order of operation. The first step is to simplify the radicand.
6  12
4
Next, simplify the square root if possible. Finally, if the numerator and denominator contain a common FACTOR, cancel it.
6  4  3
4

6  2 3
4


2 3  3
4


2 3  3

42


3  3
2
Homework
1.
In your own words, describe the order of operations in mathematics. What “level” is taking a square root?
2.
In your own words, describe the steps for simplifying an expression of the type
b c d
a
.
Simplify.
3.
3  9  4
2
4.
5  25 16
8
5.
6  36  4
2
6.
10  100 16
4
7.
7  49  1
2
8.
8  64  20
10
9.
4  16  24
4
10.
1  1  120
12
Section 9.5 — Simplifying Radical Expressions — Part II
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Math 154 :: Elementary Algebra
Section 9.6
Chapter 9 — Radical Expressions and Equations
Radical Equations
9.6 — Review of Radical Expressions and Equations Worksheet
Example:
Solve. If there is no solution, state so.
a) 4  x  3  x  5
To solve a radical equation first ISOLATE the radical expression.
4
x3  x5
Subtract 4 from both sides of the equation.
x  3  x 1
To “undo” the square rooting, square both sides of the equation.

x3

2
  x  1
2
Don’t forget to FOIL when necessary.
Continue to solve the remaining equation.
x  3   x  1  x  1 
x  3  x2  2 x  1
Subtract x and 3 from both sides of the equation.
0  x2  x  2
Since this is a quadratic equation, solve by factoring.
0   x  2   x  1
Since this is a quadratic equation, solve by factoring.
x 1  0
x20
or
 2
x
x
1
Check all solutions!
Check for x  1 :
Check for x   2 :
4
  2  3
?
  2  5
4 1 ? 3
41 ? 3
5  3
4  1 3 ? 1 5
4  4 ? 1 5
4 2 ? 6
6  6
Since –2 doesn’t work, the solution is only x  1 .
Homework
1.
2.
Multiply each of the following. If you know how to spot the difference between these problems and compute them correctly, you’ll
have an easier time solving the equations in this section.
a)
 2x 2
b)
2 x 
c)
 x  2 2
d)

2
x2

2
In your own words, describe how to recognize a square root equation. What is the inverse of square rooting?
Section 9.6 — Radical Equations
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Math 154 :: Elementary Algebra
3.
Chapter 9 — Radical Expressions and Equations
In your own words, describe how to solve a square root equation. Is it necessary to check your answers when solving a square
root equation? In your own words, explain why or why not.
Solve. Check your answers. If there is no solution, state so.
4.
x 8
5.
y 4
6.
3  a  12
7.
22  12  m
8.
2w  1  3
9.
1  4 p 11
10.
7  1 3  3x
11.
n3 5
12.
10 
q6
13.
17  k  5  23
14.
1 3  8w  3  1 2
15.
33  35  d  7
16.
4 5  2y  6  4 9
17.
3x  8 
18.
2k  5  k  1  0
19.
z 1 
20.
2 3a  5 
21.
22.
x
z5
2a
7p  38  3 p  4
4 n5 
n  35
23.
7q  3  3 q
24.
x  x 12
25.
k 8  k 6
26.
w
w  20
27.
7 
28.
1
29.
2 a  a3
30.
2 7c  8  3c
31.
2m  3 0m  1 4
32.
2  4p  5  p
33.
6  2x  3  x
y  13  y
z 1  z
Section 9.6 — Radical Equations
Caspers
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