F. Equations and Formulas Involving Radicals
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Section R.6 Radicals, Rational Exponents, and Radical Equations
EXAMPLE 14
ᮣ
Solving Radical Equations
Solve each radical equation:
a. 13x Ϫ 2 ϩ 12 ϭ x ϩ 10
Solution
ᮣ
73
3
b. 2 1
xϪ5ϩ4ϭ0
a. 13x Ϫ 2 ϩ 12 ϭ x ϩ 10
13x Ϫ 2 ϭ x Ϫ 2
1 13x Ϫ 22 2 ϭ 1x Ϫ 22 2
3x Ϫ 2 ϭ x2 Ϫ 4x ϩ 4
0 ϭ x2 Ϫ 7x ϩ 6
0 ϭ 1x Ϫ 62 1x Ϫ 12
x Ϫ 6 ϭ 0 or x Ϫ 1 ϭ 0
x ϭ 6 or x ϭ 1
Check
ᮣ
x ϭ 6:
Check
ᮣ
x ϭ 1:
original equation
isolate radical term (subtract 12)
apply power property, power is even
simplify, square binomial
set equal to zero
factor
apply zero product property
result, check for extraneous roots
13162 Ϫ 2 ϩ 12 ϭ 162 ϩ 10
116 ϩ 12 ϭ 16
16 ϭ 16 ✓
13112 Ϫ 2 ϩ 12 ϭ 112 ϩ 10
11 ϩ 12 ϭ 11
13 ϭ 11x
The only solution is x ϭ 6; x ϭ 1 is extraneous. A calculator check
is shown in the figures.
3
b. 2 1
xϪ5ϩ4ϭ0
3
1
x Ϫ 5 ϭ Ϫ2
3
1 1x Ϫ 52 3 ϭ 1Ϫ22 3
x Ϫ 5 ϭ Ϫ8
x ϭ Ϫ3
original equation
isolate radical term (subtract 4, divide by 2)
apply power property, power is odd
3
simplify: 1
x Ϫ 52 3 ϭ x Ϫ 5
solve
Substituting Ϫ3 for x in the original equation verifies it is a solution.
Now try Exercises 49 through 52
ᮣ
Sometimes squaring both sides of an equation still results in an equation with a
radical term, but often there is one fewer than before. In this case, we simply repeat the
process, as indicated by the flowchart in Figure R.7.
EXAMPLE 15
ᮣ
Solving Radical Equations
Solve the equation: 1x ϩ 15 Ϫ 1x ϩ 3 ϭ 2.
Solution
ᮣ
1x ϩ 15 Ϫ 1x ϩ 3 ϭ 2
1x ϩ 15 ϭ 1x ϩ 3 ϩ 2
1 1x ϩ 152 2 ϭ 1 1x ϩ 3 ϩ 22 2
x ϩ 15 ϭ 1x ϩ 32 ϩ 4 1x ϩ 3 ϩ 4
x ϩ 15 ϭ x ϩ 4 1x ϩ 3 ϩ 7
8 ϭ 4 1x ϩ 3
2 ϭ 1x ϩ 3
4ϭxϩ3
1ϭx
original equation
isolate one radical
power property
1A ϩ B2 2; A ϭ 1x ϩ 3, B ϭ 2
simplify
isolate radical
divide by four
power property
possible solution
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Figure R.7
Check
1x ϩ 15 Ϫ 1x ϩ 3 ϭ 2
1112 ϩ 15 Ϫ 1112 ϩ 3 ϭ 2
116 Ϫ 14 ϭ 2
4Ϫ2ϭ2
2 ϭ 2✓
ᮣ
Radical Equations
Isolate
radical term
original equation
substitute 1 for x
simplify
solution checks
Now try Exercises 53 and 54
ᮣ
Since rational exponents are so closely related to radicals, the solution process for
each is very similar. The goal is still to “undo” the radical (rational exponent) and solve
for the unknown.
Apply
power property
Does the result
contain a radical?
Power Property of Equality
YES
For real-valued expressions u and v, with positive integers m, n, and mn
in lowest terms:
If m is odd
m
and u n ϭ v,
NO
If m is even
m
and u n ϭ v 1v 7 02,
then 1u n 2 m ϭ vm
m n
Solve using
properties of equality
then 1u n 2 m ϭ Ϯv m
n
m n
n
n
n
u ϭ vm
u ϭ Ϯvm
The power property of equality basically says that if certain conditions are satisfied, both sides of an equation can be raised to any needed power.
Check results in
original equation
EXAMPLE 16
ᮣ
Solving Equations with Rational Exponents
Solve each equation:
3
a. 31x ϩ 12 4 Ϫ 9 ϭ 15
Solution
ᮣ
3
a. 31x ϩ 12 4 Ϫ 9 ϭ 15
1x ϩ 12 ϭ 8
3
4
3 1x ϩ 12 4 ϭ 8
x ϩ 1 ϭ 16
x ϭ 15
3 4
4 3
Check
ᮣ
4
3
3
4
3115 ϩ 12 Ϫ 9 ϭ 15
3116 2 Ϫ 9 ϭ 15
3122 3 Ϫ 9 ϭ 15
3182 Ϫ 9 ϭ 15
15 ϭ 15 ✓
1
4
b.
3
1x Ϫ 32 ϭ 4
2
3
3 1x Ϫ 32 4 ϭ Ϯ 4
xϪ3ϭ Ϯ8
xϭ3Ϯ8
2 3
3 2
3
2
b. 1x Ϫ 32 3 ϭ 4
2
original equation; mn ϭ 34
isolate variable term (add 9, divide by 3)
apply power property, note m is odd
simplify 383 ϭ 183 2 4 ϭ 16 4
4
1
result
substitute 15 for x in the original equation
simplify, rewrite exponent
4
1
16 ϭ 2
23 ϭ 8
solution checks
original equation; mn ϭ 23
apply power property, note m is even
simplify 342 ϭ 142 2 3 ϭ 8 4
3
1
result
The solutions are 3 ϩ 8 ϭ 11 and 3 Ϫ 8 ϭ Ϫ5.
Verify by checking both in the original equation.
Now try Exercises 55 through 58
ᮣ
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Section R.6 Radicals, Rational Exponents, and Radical Equations
ᮣ
CAUTION
As you continue solving equations with radicals and rational exponents, be careful not to
arbitrarily place the “Ϯ” sign in front of terms given in radical form. The expression 118
indicates the positive square root of 18, where 118 ϭ 312. The equation x2 ϭ 18 becomes x ϭ Ϯ 118 after applying the power property, with solutions x ϭ Ϯ312
1x ϭ Ϫ312, x ϭ 3122, since the square of either number produces 18.
In Section R.4, we used a technique called u-substitution to factor expressions
in quadratic form. The following equations are also in quadratic form
since
the de2
1
gree of the leading term is twice the degree of the middle term: x3 Ϫ 3x3 Ϫ 10 ϭ 0
and 1x2 ϩ x2 2 Ϫ 81x2 ϩ x2 ϩ 12 ϭ 0. The first equation and its solution appear in
Example 17.
EXAMPLE 17
ᮣ
Solving Equations in Quadratic Form
2
1
Solve using a u-substitution: x3 Ϫ 3x3 Ϫ 10 ϭ 0.
Solution
ᮣ
This
equation
is in quadratic form since it can be rewritten as:
1
1
1x3 2 2 Ϫ 31x3 2 1 Ϫ 10 ϭ
0, where the2 degree of leading term is twice that of second
1
term. If we let u ϭ x3, then u2 ϭ x3 and the equation becomes u2 Ϫ 3u1 Ϫ 10 ϭ 0,
which is factorable.
1u Ϫ 521u ϩ 22
uϭ5
1
x3 ϭ 5
1
1x3 2 3 ϭ 53
x ϭ 125
ϭ0
or
u ϭ Ϫ2
1
or
x3 ϭ Ϫ2
1
or 1x3 2 3 ϭ 1Ϫ22 3
or
x ϭ Ϫ8
factor
solution in terms of u
1
resubstitute x 3 for u
cube both sides: 13 132 ϭ 1
solve for x
Both solutions check.
Now try Exercises 59 and 60
Figure R.8
Hypotenuse
Leg
90Њ
Leg
ᮣ
A right triangle is one that has a 90° angle (see Figure R.8). The longest side (opposite
the right angle) is called the hypotenuse, while the other two sides are simply called
“legs.” The Pythagorean theorem is a formula that says if you add the square of each
leg, the result will be equal to the square of the hypotenuse. Furthermore, we note the
converse of this theorem is also true.
Pythagorean Theorem
1. For any right triangle with legs a and b and hypotenuse c, a2 ϩ b2 ϭ c2
2. For any triangle with sides a, b, and c, if a2 ϩ b2 ϭ c2, then the triangle
is a right triangle.
A geometric interpretation of the theorem is given in Figure R.9, which shows
Figure R.9
32 ϩ 42 ϭ 52.
Area
16 in2
4
5
ea
Ar in2
25
3
Area
9 in2
25
13
12
24
ϩ
ϭ
25 ϩ 144 ϭ 169
52
122
c
7
5
132
b
ϩ
ϭ
49 ϩ 576 ϭ 625
72
242
252
ϩ b2 ϭ c2
general case
a2
a
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EXAMPLE 18
ᮣ
Applying the Pythagorean Theorem
An extension ladder is placed 9 ft from the base of a building in an effort to reach a
third-story window that is 27 ft high. What is the minimum length of the ladder
required? Answer in exact form using radicals, and in approximate form by
rounding to one decimal place.
Solution
ᮣ
We can assume the building makes a 90° angle with the
ground, and use the Pythagorean theorem to find the
required length. Let c represent this length.
c2 ϭ a2 ϩ b2
c2 ϭ 192 2 ϩ 1272 2
c2 ϭ 81 ϩ 729
c2 ϭ 810
c ϭ 1810
c ϭ 9 110
c Ϸ 28.5 ft
Pythagorean theorem
substitute 9 for a and 27 for b
92 ϭ 81, 272 ϭ 729
c
add
27 ft
definition of square root; c 7 0
exact form: 1810 ϭ 181 # 10 ϭ 9 110
approximate form
The ladder must be at least 28.5 ft tall.
9 ft
F. You’ve just seen how
we can solve equations and
use formulas involving radicals
Now try Exercises 63 and 64
R.6 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
n
1. 1an ϭ ͿaͿ if n 7 0 is a(n)
integer.
3. By decomposing the rational exponent, we can
3
?
rewrite 16 4 as 116 ? 2 ?.
5. Discuss/Explain what it means when we say an
expression like 1A has been written in simplest
form.
ᮣ
2. The conjugate of 2 Ϫ 13 is
.
4. 1x2 2 3 ϭ x2 3 ϭ x1 is an example of the
property of exponents.
3 2
#
3 2
6. Discuss/Explain why it would be easier x12
to simplify the expression given using
1
rational exponents rather than radicals. x3
DEVELOPING YOUR SKILLS
Evaluate the expression 2x2 for the values given.
7. a. x ϭ 9
b. x ϭ Ϫ10
8. a. x ϭ 7
b. x ϭ Ϫ8
Simplify each expression, assuming that variables can
represent any real number.
9. a. 249p2
c. 281m4
b. 21x Ϫ 32 2
d. 2x2 Ϫ 6x ϩ 9
10. a. 225n2
c. 2v10
3
11. a. 1
64
3
c. 2
216z12
3
12. a. 1
Ϫ8
3
c. 2
27q9
b. 21y ϩ 22 2
d. 24a2 ϩ 12a ϩ 9
3
b. 2
Ϫ216x3
d.
v3
B Ϫ8
3
3
b. 2
Ϫ125p3
d.
w3
B Ϫ64
3
ᮣ
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Section R.6 Radicals, Rational Exponents, and Radical Equations
6
13. a. 1
64
5
c. 2243x10
5
e. 2 1k Ϫ 32 5
4
14. a. 1 81
5
c. 21024z15
5
e. 2 1q Ϫ 92 5
3
15. a. 1 Ϫ125
6
b. 1
Ϫ64
5
d. 2Ϫ243x5
6
f. 2 1h ϩ 22 6
4
b. 1 Ϫ81
5
d. 2Ϫ1024z20
6
f. 2 1p ϩ 42 6
4
b. Ϫ 281n12
10
c. 2Ϫ36
49v
B 36
4
b. Ϫ 216m24
d.
16. a. 1 Ϫ216
3
c. 2Ϫ121
25x6
d.
B 4
24. a. 28x6
2 3
c. 227a2b6
9
12 Ϫ 248
e.
8
25. a. 2.5 218a22a3
c.
26. a. 5.1 22p232p5
c.
3
16 2
b. a b
25
2
3
17. a. 8
4 Ϫ2
c. a b
25
3
d. a
27. a.
Ϫ27p6
8q3
2
3
b
28. a.
4
b. a b
9
3
16 Ϫ4
c. a b
81
3
2
Ϫ125v9 3
b
d. a
27w6
b. aϪ
3
19. a. Ϫ1442
c. 1Ϫ272 Ϫ3
2
b. aϪ
3
20. a. Ϫ1002
c. 1Ϫ1252
d. Ϫa
4
b
25
21. a. 12n p 2
22. a. a
1
4x2
b
23y
20
B 4x4
5
29. a. 2
32x10y15
Ϫ20 ϩ 232
4
2
b. Ϫ 23b212b2
3
f.
3
3
d. 29v2u23u5v2
4
b. Ϫ 25q220q3
5
d. 25cd2 125cd
3
b.
c. 31 b
4
e. 2b 1
b
27x
b
64
3
4
30. a. 2
81a12b16
4
c. 32a
3
4
e. 1
c1
c
3
2
x9 Ϫ3
d. Ϫa b
8
3
3
2
108n4
3
2
4n
81
d. 12 3 9
A 8z
b.
3
2
72b5
3
2
3b2
125
d. Ϫ9 3
A 27x6
4 5
b. x 2
x
d.
3
1
6
26
5 6
b. a 2
a
d.
3
1
3
4
2
3
4
Ϫ23
2 Ϫ25 5
3
c.
227y7
4
Use properties of exponents to simplify. Answer in
exponential form without negative exponents.
24x8
28m5
d. 254m6n8
3
2
3 Ϫ43
49
b
36
ab2 25ab4
B 3 B 27
22m
45
c.
B 16x2
3
2
18. a. 92
x3y 4x5y
B 3 B 12y
3
b. 3 2
128a4b2
2
b. a
3
8y4
3
64y2
b
1
3
b. 12x y 2
Ϫ14
3
4
4
Simplify each expression. Assume all variables represent
nonnegative real numbers.
23. a. 218m2
3 3
c. 264m3n5
8
Ϫ6 ϩ 228
e.
2
3
b. Ϫ2 2
Ϫ125p3q7
d. 232p3q6
f.
27 Ϫ 272
6
Simplify and add (if possible).
31. a.
b.
c.
d.
12 272 Ϫ 9 298
8 248 Ϫ 3 2108
7 218m Ϫ 250m
2 228p Ϫ 3 263p
32. a.
b.
c.
d.
Ϫ3280 ϩ 2 2125
5 212 ϩ 2 227
3 212x Ϫ 5 275x
3 240q ϩ 9 210q
3
3
33. a. 3x 1 54x Ϫ 5 216x4
b. 14 ϩ 13x Ϫ 112x ϩ 145
c. 272x3 ϩ 150 Ϫ 17x ϩ 127
3
3
34. a. 5 254m3 Ϫ 2m216m3
b. 110b ϩ 1200b Ϫ 120 ϩ 140
c. 275r3 ϩ 132 Ϫ 127r ϩ 138
77
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Compute each product and simplify the result.
35. a. 17 122
b. 131 15 ϩ 172
c. 1n ϩ 1521n Ϫ 152 d. 16 Ϫ 132 2
46. a.
7
17 ϩ 3
b.
12
1x ϩ 13
47. a.
110 Ϫ 3
13 ϩ 12
b.
7 ϩ 16
3 Ϫ 3 12
37. a. 13 ϩ 217213 Ϫ 2 172
b. 1 15 Ϫ 1142 1 12 ϩ 1132
c. 12 12 ϩ 6 162 13110 ϩ 172
48. a.
1 ϩ 12
16 ϩ 114
b.
1 ϩ 16
5 ϩ 2 13
2
36. a. 10.3 152 2
b. 151 16 Ϫ 122
c. 14 ϩ 13214 Ϫ 132 d. 12 ϩ 152 2
38. a. 15 ϩ 4110211 Ϫ 2 1102
b. 1 13 ϩ 122 1 110 ϩ 1112
c. 1315 ϩ 4 122 1 115 ϩ 162
Use a substitution to verify the solutions to the
quadratic equation given. Verify results using a
calculator.
39. x2 Ϫ 4x ϩ 1 ϭ 0
a. x ϭ 2 ϩ 13
b. x ϭ 2 Ϫ 13
40. x Ϫ 10x ϩ 18 ϭ 0
a. x ϭ 5 Ϫ 17
b. x ϭ 5 ϩ 17
41. x2 ϩ 2x Ϫ 9 ϭ 0
a. x ϭ Ϫ1 ϩ 110
b. x ϭ Ϫ1 Ϫ 110
42. x2 Ϫ 14x ϩ 29 ϭ 0
a. x ϭ 7 Ϫ 2 15
b. x ϭ 7 ϩ 2 15
2
Rationalize each expression by building perfect nth root
factors for each denominator. Assume all variables
represent positive quantities.
3
112
27
c.
B 50b
5
e. 3
1a
43. a.
20
B 27x3
1
d. 3
A 4p
b.
Ϫ4
125
44. a.
b.
B 12n3
120
5
3
c.
d. 3
B 12x
A 2m2
Ϫ8
e. 3
3 15
Simplify the following expressions by rationalizing the
denominators. Where possible, state results in exact
form and approximate form, rounded to hundredths.
45. a.
8
3 ϩ 111
b.
6
1x Ϫ 12
Solve each equation and check your solutions by
substitution. Identify any extraneous roots.
49. a. Ϫ313x Ϫ 5 ϭ Ϫ9
b. x ϭ 13x ϩ 1 ϩ 3
50. a. Ϫ2 14x Ϫ 1 ϭ Ϫ10
b. Ϫ5 ϭ 15x Ϫ 1 Ϫ x
3
51. a. 2 ϭ 13m Ϫ 1
3
b. 2 1 7 Ϫ 3x Ϫ 3 ϭ Ϫ7
3
1 2m ϩ 3
ϩ2ϭ3
c.
Ϫ5
3
3
d. 1 2x Ϫ 9 ϭ 1 3x ϩ 7
3
52. a. Ϫ3 ϭ 1 5p ϩ 2
3
b. 3 1 3 Ϫ 4x Ϫ 7 ϭ Ϫ4
3
1
6x Ϫ 7
Ϫ 5 ϭ Ϫ6
c.
4
3
3
d. 3 1 x ϩ 3 ϭ 2 1 2x ϩ 17
53. a.
b.
c.
d.
1x Ϫ 9 ϩ 1x ϭ 9
x ϭ 3 ϩ 223 Ϫ x
1x Ϫ 2 Ϫ 12x ϭ Ϫ2
112x ϩ 9 Ϫ 124x ϭ Ϫ3
54. a.
b.
c.
d.
1x ϩ 7 Ϫ 1x ϭ 1
12x ϩ 31 ϩ x ϭ 2
13x ϭ 1x Ϫ 3 ϩ 3
13x ϩ 4 Ϫ 17x ϭ Ϫ2
Write the equation in simplified form, then solve. Check
all answers by substitution.
3
55. a. x5 ϩ 17 ϭ 9
3
b. Ϫ2x4 ϩ 47 ϭ Ϫ7
5
56. a. 0.3x2 Ϫ 39 ϭ 42
5
b. 0.5x3 ϩ 92 ϭ Ϫ43
2
57. a. 21x ϩ 52 3 Ϫ 11 ϭ 7
4
b. Ϫ31x Ϫ 22 5 ϩ 29 ϭ Ϫ19
1
3
58. a. 3x3 Ϫ 10 ϭ 1
x
2
5 2
b. 2 1x Ϫ 4 ϭ x5
2
1
59. x3 Ϫ 2x3 Ϫ 15 ϭ 0
3
60. x3 Ϫ 9x2 ϭ Ϫ8
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Section R.6 Radicals, Rational Exponents, and Radical Equations
WORKING WITH FORMULAS
1
61. Fish length to weight relationship: L ؍1.131W2 3
The length to weight relationship of a female
Pacific halibut can be approximated by the formula
shown, where W is the weight in pounds and L is
the length in feet. A fisherman lands a halibut that
weighs 400 lb. Approximate the length of the fish
(round to two decimal places).
ᮣ
79
1s
4
The time it takes an object to fall a certain distance is
given by the formula shown, where t is the time in
seconds and s is the distance the object has fallen.
Approximate the time it takes an object to hit the
ground, if it is dropped from the top of a building
that is 80 ft in height (round to hundredths).
62. Timing a falling object: t ؍
APPLICATIONS
63. Length of a cable: A radio
tower is secured by cables that
are anchored in the ground 8 m
from its base. If the cables are
attached to the tower 24 m above
the ground, what is the length of
each cable? Answer in (a) exact
form using radicals, and (b)
approximate form by rounding
to one decimal place.
24 m
c
8m
64. Height of a kite: Benjamin
Franklin is flying his kite in a storm once again.
John Adams has walked to a position directly under
the kite and is 75 ft from Ben. If the kite is 50 ft
above John Adams’ head, how much string S has
Ben let out? Answer in (a) exact form using
radicals, and (b) approximate form by rounding to
one decimal place.
S
50 ft
75 ft
The time T (in days) required for a planet to
make one revolution around3 the sun is modeled
by the function T ؍0.407R2, where R is the
maximum radius of the planet’s orbit (in
millions of miles). This is known as Kepler’s
third law of planetary motion. Use the equation
given to approximate the number of days
required for one complete orbit of each planet,
given its maximum orbital radius.
65. a. Earth: 93 million mi
b. Mars: 142 million mi
c. Mercury: 36 million mi
66. a. Venus: 67 million mi
b. Jupiter: 480 million mi
c. Saturn: 890 million mi
67. Accident investigation: After an accident, police
officers will try to determine the approximate
velocity V that a car was traveling using the formula
V ϭ 2 26L, where L is the length of the skid marks
in feet and V is the velocity in miles per hour. (a) If
the skid marks were 54 ft long, how fast was the car
traveling? (b) Approximate the speed of the car if
the skid marks were 90 ft long.
68. Wind-powered energy: If a wind-powered
generator is delivering P units of power, the
velocity V of the wind (in miles per hour) can be
3 P
, where k is a constant
determined using V ϭ
Ak
that depends on the size and efficiency of the
generator. Rationalize the radical expression and
use the new version to find the velocity of the wind
if k ϭ 0.004 and the generator is putting out 13.5
units of power.
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69. Surface area: The lateral surface
area (surface area excluding the base)
h
S of a cone is given by the formula
2
2
S ϭ r 2r ϩ h , where r is the
r
radius of the base and h is the height
of the cone. Find the lateral surface
area of a cone that has a radius of 6 m and a height of
10 m. Answer in simplest form.
70. Surface area: The lateral surface
a
area S of a frustum (a truncated
cone) is given by the formula
h
S ϭ 1a ϩ b2 2h2 ϩ 1b Ϫ a2 2,
b
where a is the radius of the upper
base, b is the radius of the lower
base, and h is the height. Find the surface area of a
frustum where a ϭ 6 m, b ϭ 8 m, and h ϭ 10 m.
Answer in simplest form.
71. Planetary motion: The time T (in days) for a
planet to make one revolution (elliptical orbit)
3
around the sun is modeled by T ϭ 0.407R2, where
R is the maximum radius of the planet’s orbit in
ᮣ
millions of miles (Kepler’s third law of planetary
motion). Use the equation to approximate the
maximum radius of each orbit, given the number of
days it takes for one revolution. (See Exercises 65
and 66.)
a. Mercury: 88 days
b. Venus: 225 days
c. Earth: 365 days
d. Mars: 687 days
e. Jupiter: 4333 days
f. Saturn: 10,759 days
72. Wind-powered energy: If a wind-powered
generator is delivering P units of power, the
velocity V of the wind (in miles per hour) can be
3 P
, where k is a constant
determined using V ϭ
Ak
that depends on the size and efficiency of the
generator. Given k ϭ 0.004, approximately how
many units of power are being delivered if the
wind is blowing at 27 miles per hour? (See
Exercise 68.)
EXTENDING THE CONCEPT
The expression x2 ؊ 7 is not factorable using integer
values. But the expression can be written in the form
x2 ؊ 1 272 2, enabling us to factor it as a “binomial”
and its conjugate: 1x ؉ 272 1x ؊ 272. Use this idea to
factor the following expressions.
73. a. x2 Ϫ 5
b. n2 Ϫ 19
74. a. 4v2 Ϫ 11
b. 9w2 Ϫ 17
75. The following terms form a pattern that continues
until the sixth term is found:
23x ϩ 29x ϩ 227x ϩ p (a) Compute the sum
of all six terms; (b) develop a system (investigate
the pattern further) that will enable you to find the
sum of 12 such terms without actually writing out
the terms.
76. Find a quick way to simplify the expression
without the aid of a calculator.
a a a a a3 b
5
6
3
2
4
5
3
4
2
5
aaaa
80
10
3
1
1
1
1
9
77. If 1x2 ϩ xϪ2 2 2 ϭ , find the value of x2 ϩ xϪ2.
2
78. Rewrite by rationalizing the numerator:
1x ϩ h Ϫ 1x
h
Determine the values of x for which each expression
represents a real number.
79.
1x Ϫ 1
x2 Ϫ 4
80.
x2 Ϫ 4
1x Ϫ 1
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Overview of Chapter R
OVERVIEW OF CHAPTER R
Prerequisite Definitions, Properties, Formulas, and Relationships
Notation and Relations
concept
• Set notation:
notation
{members}
ʦ
• Is an element of
л or { }
• Empty set
Is
a
proper
subset
of
(
•
5x | x p6
• Defining a set
description
braces enclose the members of a set
indicates membership in a set
a set having no elements
indicates the elements of one set
are entirely contained in another
the set of all x, such that x . . .
Sets of Numbers
• Natural: ގϭ 51, 2, 3, 4, p6
• Integers: ޚϭ 5p , Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3, p6
• Irrational: ވϭ {numbers with a nonterminating,
example
set of even whole numbers
A ϭ 50, 2, 4, 6, 8, p6
14 ʦ A
odd numbers in A
S ϭ 50, 6, 12, 18, 24, p6
S ( A
S ϭ 5x | x ϭ 6n for n ʦ ޗ6
• Whole: ޗϭ 50, 1, 2, 3, p6
• Rational: ޑϭ e q , where p, q ʦ ;ޚq 0 f
• Real: ޒϭ {all rational and irrational numbers}
p
nonrepeating decimal form}
Absolute Value of a Number
Distance between numbers a and b
on the number line
• The distance between a number n
Ϳa Ϫ bͿ or Ϳb Ϫ aͿ
and zero (always positive)
n
if n Ն 0
0n 0 ϭ e
Ϫn if n 6 0
For a complete review of these ideas, go to www.mhhe.com/coburn.
R.1 Properties of Real Numbers: For real numbers a, b, and c,
Commutative Property
• Addition: a ϩ b ϭ b ϩ a
• Multiplication: a # b ϭ b # a
Identities
Associative Property
• Addition: 1a ϩ b2 ϩ c ϭ a ϩ 1b ϩ c2
• Multiplication: 1a # b2 # c ϭ a # 1b # c2
Inverses
• Additive: a ϩ 1Ϫa2 ϭ 0
• Additive: 0 ϩ a ϭ a
• Multiplicative: 1 # a ϭ a
p q
• Multiplicative: q # p ϭ 1; p, q
0
R.2 Properties of Exponents: For real numbers a and b, and integers m, n, and p
•
•
•
•
(excluding 0 raised to a nonpositive power),
Product property: b
• Power property: 1bm 2 n ϭ bmn
am p amp
Product to a power: 1ambn 2 p ϭ amp # bnp
Quotient
to
a
power:
b ϭ np 1b 02
a
•
bm
bn
b
mϪn
Quotient property: n ϭ b
1b 02
0
Zero
exponents:
ϭ
1
1b
02
b
b
•
Ϫn
n
1
a
b
• Scientific notation: N ϫ 10k; 1 Յ ͿNͿ 6 10, k ʦ ޚ
Negative exponents: bϪn ϭ n ; a b ϭ a b
a
b
b
1a, b 02
m
# bn ϭ bmϩn
81
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CHAPTER R A Review of Basic Concepts and Skills
R–82
R.2 Polynomials
• A polynomial is a sum or difference of monomial terms
• Polynomials are classified as a monomial, binomial, trinomial, or polynomial, depending on the number of terms
• The degree of a polynomial in one variable is the same as the largest exponent occuring on the variable in any term
• A polynomial expression is in standard form when written with the terms in descending order of degree
R.3 Solving Linear Equations and Inequalities
• Properties of Equality:
additive property
If A and B are algebraic expressions, where A ϭ B,
then A ϩ C ϭ B ϩ C.
•
•
•
•
•
•
•
•
multiplicative property
If A and B are algebraic expressions, where A ϭ B,
then A # C ϭ B # C
B
A
[C can be positive or negative]
and ϭ ;C 0
C
C
A linear equation in one variable is one that can be written in the form ax ϩ b ϭ c, where the exponent on the
variable is a 1.
To solve a linear equation, we attempt to isolate the term containing the variable using the additive property, then
solve for the variable using the multiplicative property.
An equation can be an identity (always true), a contradiction (never true) or conditional (true or false depending
on the input value[s]).
If an equation contains fractions, multiplying both sides by the least common denominator of all fractions will
“clear the denominators” and reduce the amount of work required to solve.
Inequalities are solved using properties similar to those used for solving equations. The one exception is when
multiplying or dividing by a negative quantity, as the inequality symbol must then be reversed to maintain the
truth of the resulting statement.
Solutions to an inequality can be given using a simple inequality, graphed on a number line, stated in set notation,
or stated using interval notation.
Given two sets A and B: A intersect B 1A ă B2 is the set of elements shared by both A and B (elements common to
both sets). A union B 1A ´ B2 is the set of elements in either A or B (elements are combined to form a larger set).
Compound inequalities are formed using the conjunction “and” or the conjunction “or.” The result can be either a
joint inequality as in Ϫ3 6 x Յ 5, or a disjoint inequality, x 6 Ϫ2 or x 7 7.
R.4 Special Factorizations
• A2 Ϫ B2 ϭ 1A ϩ B21A Ϫ B2
• A2 Ϯ 2AB ϩ B2 ϭ 1A Ϯ B2 2
3
3
2
2
• A Ϫ B ϭ 1A Ϫ B21A ϩ AB ϩ B 2
• A3 ϩ B3 ϭ 1A ϩ B21A2 Ϫ AB ϩ B2 2
• Certain equations of higher degree can be solved using factoring skills and the zero product property.
R.5 Rational Expressions: For polynomials P, Q, R, and S with no denominator of zero,
P#R
P
P#R
P
ϭ
Equivalence:
ϭ
•
Q#R
Q
Q
Q#R
P R
P#R
PR
R
P S
PS
P
Multiplication: # ϭ # ϭ
• Division: Ϭ ϭ # ϭ
Q S
Q S
QS
Q
S
Q R
QR
Q
PϩQ
Q
PϪQ
P
P
Addition: ϩ ϭ
• Subtraction: Ϫ ϭ
R
R
R
R
R
R
Addition/subtraction with unlike denominators:
1. Find the LCD of all rational expressions.
2. Build equivalent expressions using LCD.
3. Add/subtract numerators as indicated.
4. Write the result in lowest terms.
To solve rational equations, first clear denominators using the LCD, noting values that must be excluded.
Multiplying an equation by a variable quantity sometimes introduces extraneous solutions. Check all results in the
original equation.
• Lowest terms:
•
•
•
•
•
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Practice Test
R.6 Properties of Radicals
• 1a is a real number only for a Ն 0
n
• 1a ϭ b, only if bn ϭ a
83
• 1a ϭ b, only if b2 ϭ a
n
• If n is even, 1a represents a real number only
if a Ն 0
n
• For any real number a, 1an ϭ a when n is odd
m
If a is a real number and
n is an integer greater
• If n is a rational number written in lowest terms
1
n
n
m
m
than 1, then 1 a ϭ an provided 1 a represents a
n
n
with n Ն 2, then a n ϭ 11a2 m and a n ϭ 1am
real number
n
provided 2a represents a real number.
n
n
n
n
If 1A and 1B represent real numbers and B 0,
If 1A and 1B represent real numbers,
•
n
n
n
n
1AB ϭ 1 A # 1B
1A
n A
ϭ n
BB
1B
A radical expression is in simplest form when:
1. the radicand has no factors that are perfect nth roots,
2. the radicand contains no fractions, and
3. no radicals occur in a denominator.
To solve radical equations, isolate the radical on one side, then apply the appropriate “nth power” to free up the
radicand. Repeat the process if needed. See flowchart on page 74.
For equations with a rational exponent mn, isolate the variable term and raise both sides to the mn power. If m is even,
there will be two real solutions.
n
• For any real number a, 1an ϭ ͿaͿ when n is even
•
•
•
•
•
R.6 Pythagorean Theorem
• For any right triangle with legs a and b and
hypotenuse c: a2 ϩ b2 ϭ c2.
• For any triangle with sides a, b, and c, if
a2 ϩ b2 ϭ c2, then the triangle is a right triangle.
PRACTICE TEST
1. State true or false. If false, state why.
7
a. 13 ϩ 42 2 ϭ 25
b. ϭ 0
0
c. x Ϫ 3 ϭ Ϫ3 ϩ x
d. Ϫ21x Ϫ 32 ϭ Ϫ2x Ϫ 3
2. State the value of each expression.
3
a. 1 121
b. 1
Ϫ125
c. 1Ϫ36
d. 1400
3. Evaluate each expression:
7
1
5
1
a. Ϫ aϪ b
b. Ϫ Ϫ
8
4
3
6
c. Ϫ0.7 ϩ 1.2
d. 1.3 ϩ 1Ϫ5.92
4. Evaluate each expression:
1
a. 1Ϫ42aϪ2 b
b. 1Ϫ0.621Ϫ1.52
3
Ϫ2.8
c.
d. 4.2 Ϭ 1Ϫ0.62
Ϫ0.7
#
0.08 12 10
5. Evaluate using a calculator: 2000 a1 ϩ
b
12
6. State the value of each expression, if possible.
a. 0 Ϭ 6
b. 6 Ϭ 0
7. State the number of terms in each expression and
identify the coefficient of each.
cϩ2
ϩc
a. Ϫ2v2 ϩ 6v ϩ 5
b.
3
8. Evaluate each expression given x ϭ Ϫ0.5 and
y ϭ Ϫ2. Round to hundredths as needed.
y
a. 2x Ϫ 3y2
b. 12 Ϫ x14 Ϫ x2 2 ϩ
x
9. Translate each phrase into an algebraic expression.
a. Nine less than twice a number is subtracted from
the number cubed.
b. Three times the square of half a number is
subtracted from twice the number.
10. Create a mathematical model using descriptive
variables.
a. The radius of the planet Jupiter is approximately
119 mi less than 11 times the radius of the Earth.
Express the radius of Jupiter in terms of the
Earth’s radius.
b. Last year, Video Venue Inc. earned $1.2 million
more than four times what it earned this year.
Express last year’s earnings of Video Venue Inc.
in terms of this year’s earnings.